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Maximal s-Wise t-Intersecting Families of Sets: Kernels, Generating Sets, and Enumeration
Journal of Combinatorial Theory, Series A 87, 5273 (1999) Article ID jcta.1998.2946, available online at http:www.idealibrary.com on
Maximal s-Wise t-Intersecting Families of Sets: Kernels, Generating Sets, and Enumeration Lucia Moura Department of Computer Science, University of Toronto, Toronto, Canada, M5S 3G4 E-mail: luciacs. toronto.edu Communicated by the Managing Editors Received July 28, 1997
A family A of k-subsets of an n-set is said to be s-wise t-intersecting if |A 1 & } } } & A s | t, for any A 1 , ..., A s # A. For fixed s, n, k, and t, let I s(n, k, t) denote the set of all such families. A family A # I s(n, k, t) is said to be maximal if it is not properly contained in any other family in I s(n, k, t). We show that for fixed s, k, t, there is an integer n0 =n 0 (k, s, t), for which the maximal families in I s(n 0 , k, t) completely determine the maximal families in I s(n, k, t), for all n n 0 . We give a construction for maximal families in I s(n+1, k+1, t+1) based on those in I s(n, k, t). Finally, for s=2, we classify the maximal families for k=t+1, n t+2, t 1, and for k=t+2, n t+6, t 1. The concepts of kernels and generating sets of a family of subsets play an important role in this work. 1999 Academic Press, Inc.
1. MOTIVATION, DEFINITIONS, AND RESULTS For i, j # N, we abbreviate the set [i, i+1, ..., j] as [i, j]. Given s 2 and n, t 1, a family A of subsets of [1, n] is said to be s-wise t-intersecting, if any s members A 1 , ..., A s of A are such that |A 1 & } } } & A s | t. A family A is said to be k-uniform if every member of A has cardinality k. We denote by I s(n, k, t) the set of all k-uniform s-wise t-intersecting families of subsets of [1, n]. We are interested in the families in I s(n, k, t) that are maximal with respect to set inclusion, i.e., the families A # I s(n, k, t) such that for any B # I s(n, k, t), if B$A then B=A. Throughout this paper, we denote by MI s(n, k, t) the set of all families in I s(n, k, t) that are maximal in this sense. The problem that motivated this paper is the following. Problem 1. Given s, n, k, and t, list all families of MI s(n, k, t); or alternatively, list all non isomorphic families of MI s(n, k, t). 52 0097-316599 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved.
MAXIMAL INTERSECTING SET FAMILIES
53
We should make clear what we mean by isomorphic families. Definition 1.1. We say that two families A 1 and A 2 of subsets of [1, n] are isomorphic, denoted by A 1 tA 2, if there exist a permutation ? of the set [1, n], such that ? sends A 1 to A 2. We also write A 2 =?(A 1 ). Problem 1 is relevant for the study of some polyhedra arising in combinatorial design theory. The families in MI 2(n, k, t) give rise to facets of the polyhedron for the t-(n, k, 1) packing designs, and are also related to the polyhedron for the t-(n, k, 1) designs, also called S(t, k, n) Steiner systems. Precise definitions of these structures as well as of their corresponding polyhedra can be found in [15]. Problem 1 is also of independent interest in extremal set theory. Related works dealing with the determination of the largest family in MI s(n, k, t) include [1, 2, 6, 9, 12]. We make use of two important concepts in the study of families in MI s(n, k, t), namely kernels and generating sets. First, we define the notion of a generating set of a family of subsets, as introduced by Ahlswede and Khachatrian [1, 2]. Let us fix some notation. For any B [1, n], let U kn (B)=[C [1, n] : C$B, |C| =k], and for a family B of subsets of [1, n], let U kn (B)= B # B U kn (B). Let A be a k-uniform family of subsets of [1, n]; a family g of subsets of [1, n] is called a generating set of A, if U kn ( g)=A. The set A itself is a generating set of A, although not a very interesting one. We would like to be able to find generating sets g that are ``minimal'' in some sense, for example, the ones for which | G # g G | or | g | are minimal. Such generating sets provide us with a compact way to represent families of subsets, which proved to be useful for studying Problem 1. Now, we turn our attention to the concept of a kernel of an s-wise t-intersecting family of subsets. For a family A in I s(n, k, t), a set K [1, n] is a kernel of A if any s sets in A meet in at least t elements of K, i.e., for any A 1 , ..., A s # A we have |A 1 & } } } & A s & K| t. Denote by n(k, s, t) the smallest integer for which every family A in n I s(n, k, t) has a kernel K(A) with |K(A)| n(k, s, t) (the standard definition of n(k, s, t) does not require the families A to be k-uniform, but rather having members with cardinality at most k; both definitions turn out to be equivalent). It is not immediate to conclude that n(k, s, t) is finite for all k, s and t. Ca*czynska-Kar*owicz [4] proved that n(k, 2, 1) is finite. The first explicit upper bound on n(k, 2, 1) was given by Ehrenfeucht and Mycielski [5]. Lower bounds and further improvements on upper bound were given by Erdo s and Lovasz [7] and Tuza [16]. The finiteness of n(k, s, t) was independently proven by Frankl [8], in an implicit form, Kahn and Seymour [13], and Frankl and Furedi [10]. These results imply that at least t elements of each of the s-wise intersections of a family A # I s(n, k, t) are included in a small subset (with cardinality independent of n) of the underlying
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LUCIA MOURA
set [1, n]. General bounds on n(k, s, t) are given by Alon and Furedi [3], who proved that 1 13 (k&t)(s&1) k&t+1+w(k&t)(s&1)x n(k, s, t) k (3 ) 3 w(k&t)(s&1)x k
\
\ +
ss (s&1) s&1
+
w(k&t)(s&1)x
.
(1)
They give an explicit construction for the lower bound, and also show that the upper bound follows from a theorem of Furedi [11]. Kohayakawa [14] sharpened the lower bound in (1). We note that the upper bound in (1) can be improved by a simple observation on the last step of Alon and Furedi's proof. Indeed, in that proof [3, Theorem 2.2], they obtain an upper bound on the maximum size of a kernel given by the size of the union of m sets of size k, with m=
\
k&t+1+w(k&t)(s&1)x . w(k&t)(s&1)x
+
This leads to the upper bound in (1). However, the m sets form an s-wise t-intersecting family, which implies that every set has at least t elements in common with a fixed set, say the first one. Therefore, the upper bound can be improved to k&t+1+w(k&t)(s&1)x +t w(k&t)(s&1)x
\ s (k&t) \(s&1) +
n(k, s, t) (k&t)
s
s&1
+
w(k&t)(s&1)x
+t.
(2)
Before summarizing the contributions in this paper, we fix some notation and give some definition. Notation 1.2. Let C be a collection of families of subsets. We denote by D(C) the number of non isomorphic members of C, that is, the number of equivalent classes induced by the equivalent relation given by t. Definition 1.3. Let C 1, C 2 be collections of families of subsets. We say that collections C 1 and C 2 are isomorphism-equivalent, if for every family A 1 # C 1 there exists a family A 2 # C 2 such that A 2 tA 1, and vice versa.
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MAXIMAL INTERSECTING SET FAMILIES
In Section 2, we study the relationship between kernels and generating sets of families in MI s(n, k, t). We introduce the notion of a set of essential elements E(A) [1, n] for a family A # I s(n, k, t). We show that for maximal families the concept of kernels, generating sets, and essential elements are closely related (Theorem 2.4). In particular, for n>2k&t and A # MI s(n, k, t), we exhibit generating sets g(A) and g$(A) of A such that E(A)= G # g(A) G= G # g$(A) G is the unique kernel of A with minimum cardinality. We also show that for n large enough, more precisely, for n n 0 (k, s, t)=max[2k&t+1, n(k, s, t)], the set formed by the generating sets g$(A) for all A # MI s(n, k, t) is isomorphism-equivalent to the set formed by the generating sets g$(B) for all B # MI s(n 0 (k, s, t), k, t). Therefore, TABLE I All Generating Sets for s=2 and k=t+2 Generating sets for all non-isomorphic families in MI 2(n, 3, 1), for n 7 Gi
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
[1] [1, 2] [1, 2] [1, 2] [1, 2] [1, 2] [1, 2] [1, 2, 3] [1, 2, 3] [1, 2, 3] [1, 2, 3] [1, 2, 3] [1, 2, 4] [1, 2, 3] [1, 2, 3]
[1, 3] [1, 3] [1, 3] [1, 3, 4] [1, 3, 4] [1, 3, 4] [1, 4, 5] [1, 2, 4] [1, 2, 4] [1, 2, 4] [1, 2, 4] [1, 2, 5] [1, 2, 4] [1, 2, 4]
[2, 3] [1, 4] [1, 4, 5] [1, 3, 5] [1, 3, 6] [1, 3, 6] [1, 6, 7] [1, 2, 5] [1, 2, 5] [1, 2, 5] [1, 2, 5] [1, 2, 6] [1, 3, 6] [1, 2, 5]
[2, 3, 4] [2, 3, 4] [1, 4, 5] [1, 4, 5] [1, 4, 5] [2, 4, 6] [1, 3, 4] [1, 3, 4] [1, 3, 4] [1, 3, 5] [1, 3, 4] [1, 4, 5] [1, 3, 4]
[2, 3, 5] [2, 3, 4] [1, 5, 6] [2, 3, 4] [2, 5, 7] [1, 3, 5] [1, 3, 6] [1, 3, 6] [1, 3, 6] [1, 3, 5] [1, 5, 6] [1, 3, 6]
[2, 3, 5] [2, 3, 5] [2, 3, 5] [3, 4, 7] [1, 4, 5] [1, 4, 5] [1, 4, 5] [1, 4, 6] [1, 3, 6] [2, 3, 5] [1, 4, 7]
[2, 4, 5] [2, 4, 6] [2, 4, 6] [3, 5, 6] [2, 3, 4] [1, 5, 6] [2, 3, 4] [2, 3, 4] [2, 3, 4] [2, 4, 6] [2, 3, 4]
[2, 3, 5] [2, 3, 5] [2, 3, 5] [2, 3, 6] [2, 3, 5] [2, 5, 6] [2, 3, 7]
[2, 4, 5] [2, 4, 6] [2, 4, 6] [2, 5, 6] [2, 3, 6] [3, 4, 5] [2, 4, 6]
[3, 4, 5] [3, 4, 5] [3, 4, 5] [3, 4, 5] [4, 5, 6] [3, 4, 6] [3, 4, 5]
Generating sets for all non-isomorphic families in MI 2(n, 4, 2), for n 8 Hi
i
One application of Construction 3.1 to G i
1 ... 15 16 17
[1, 2, 7] [1, 2, 3, 7]
[1, 3, 4, 7] [1, 4, 5, 7]
[1, 5, 6, 7] [2, 4, 6, 7]
[2, 3, 6, 7] [3, 5, 6, 7]
[2, 4, 5, 7] [1, 2, 5, 6]
[1, 2, 3, 5] [1, 3, 4, 6]
[1, 2, 4, 6] [2, 3, 4, 5]
Generating sets for all non-isomorphic families in MI 2(n, t+2, t), for t 3, n t+6 i 1 ... 17
Li (t&2) applications of Construction 3.1 to H i
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LUCIA MOURA
given k, s and t, in order to solve Problem 1 for n n 0 (k, s, t), it is enough to list the generating sets g$ for the solutions obtained for n=n 0 (k, s, t). In Section 3, we give a general construction for families in MI s(n+1, k+1, t+1) based on families in MI s(n, k, t). This construction produces non-isomorphic families when applied to non-isomorphic ones. This implies that D(MI s(n+l, k+l, t+l)) D(MI s(n, k, t)), for all l 0. In Section 4, we solve Problem 1 for s=2 and k t+2. The case of k=t+1 is simple, but is included for the sake of completeness. The solution of Problem 1 for k=t+2 is more interesting. We show that the construction given in Section 3, when applied to all sets in MI 2(n, t+2, t), gives all the sets in MI 2(n+1, t+3, t+1), for all n and t 2. This reduces the case k=t+2 to the determination of families in MI 2(n 0 (4, 2, 2), 4, 2) and MI 2(n 0 (3, 2, 1), 3, 1), which were obtained by computer. The upper bound in (2) gives n(t+1, 2, t) t+3 and n(t+2, 2, t) t+20, but we prove that n(t+1, 2, t)=t+2 and n(t+2, 2, t)=t+6, for any t 1. Moreover, the results regarding enumeration of non-isomorphic sets for the case of k&t 2 are as follows. For any t 1 we have D(MI 2(n, t+1, t))=
2,
{1,
for n t+2, for n
The form of such sets are given in Theorem 4.1. For any n t+6, we have D(MI 2(n, t+2, t))=
15,
{17,
for t=1, for t 2.
The generating sets for the corresponding families are given in Table I.
2. KERNELS AND GENERATING SETS We begin this section by introducing the new concept of a number in [1, n] being essential for a family in I s(n, k, t). Then, in Theorem 2.4, we show some properties relating kernels, generating sets, and essential numbers for families in MI s(n, k, t). In Corollary 2.5, we rewrite n(k, s, t) based on essential numbers for families in MI s(n, k, t) with n>2k&t. Then, we exhibit some special generating sets (given in (5)) for families in MI s(n, k, t) with n n 0 (k, s, t)=max[2k&t+1, n(k, s, t)], that allows us to show that a solution for Problem 1 for k, s, t and n n 0 (k, s, t) can be obtained from the solution for Problem 1 for k, s, t and n=n 0 (k, s, t) (Corollary 2.10).
MAXIMAL INTERSECTING SET FAMILIES
57
Definition 2.1. Let A # I s(n, k, t). We say that e # [1, n] is essential for A, if there exist sets A 1 , ..., A s with e # A 1 & } } } & A s such that |A 1 & } } } & A s | =t. The s-tuple (A 1 , ..., A s ) is called an essential s-tuple for e in A. We also say that e is essential for A j relatively to A, for any 1 j s. The essential set of A is defined by E(A)=[e # [1, n] : e is essential for A]. It is clear from the above definition that every kernel of a family A # I s(n, k, t) must contain E(A). In Theorem 2.4, we show that for a family A # MI s(n, k, t) with n>2k&t, the set E(A) is the unique kernel of A with minimum cardinality. We also exhibit a generating set g(A) of A that is formed by elements in E(A), and that is largely used in the rest of the paper. For proving Theorem 2.4, we need the following two lemmas. The first one generalizes a known result for s=2 (see [2]). Lemma 2.2. Let n>2k&t. If A is a family in I s(n, k, t) and g is a generating set of A, then g is an s-wise t-intersecting family. Proof. Suppose by contradiction that g is not s-wise t-intersecting. Then there exist sets G 1 , ..., G s # g such that |G 1 & } } } & G s |
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LUCIA MOURA
Proof. We will show that B # A by induction on |B"A|. If |B"A| =0, trivially B=A # A. If |B "A| 1, then choose x # B" A and y # A "B. Let B$=(B" [x]) _ [ y]. Since B$$SA and |B$"A| <|B" A| , by induction hypothesis B$ # A. Since y is not essential, B$" [ y] is t-intersecting with every (s&1)-tuple of sets in A, and so is B. Therefore, since A is maximal, it follows that B # A. K The following theorem establishes connections between kernels, generating sets and essential numbers for a family A in MI s(n, k, t). In particular, we introduce a generating set formed by essential elements of A, and we show that when n>2k&t the essential set of A is the unique kernel of A with minimum cardinality. We observe that the assumption n>2k&t is not so restrictive. For example, when s=2, if n 2k&t, then the family formed by every k-subset of [1, n] is pairwise t-intersecting, and therefore is the only family in MI 2(n, k, t). Theorem 2.4. Let A be a family in MI s(n, k, t). (1)
Let us denote by g(A), the family given by g(A)=[A & E(A) : A # A].
(3)
Then, g(A) is a generating set for A. (2)
For any kernel K of A we have that E(A) K.
(3)
If n>2k&t, then:
(a)
for any generating set g of A, the set G # g G is a kernel of A;
(b) the set E(A) is a kernel of A; moreover, E(A) is the unique kernel of A with minimum cardinality. Proof. Part (1). Trivially A U kn (g(A)), so it remains to prove that U (g(A)) A. Take B # U kn (g(A)). Then, B=S _ L with S # g(A) and L [1, n] "S. Since S # g(A), then there exists an A # A with S=A & E(A). Therefore, since B$S=A & E(A), by Lemma 2.3, B # A. k n
Part (2). Let e # E(A) and K any kernel of A. Since e is essential for A there exist sets A 1 , ..., A s # A such that e # A 1 & } } } & A s and |A 1 & } } } A s | =t. In addition, since K is a kernel of A, |K & A1 & } } } & A s | t and so e # K. Part (3)(a). Let g be a generating set of A. Since n 2k&t, by Lemma 2.2, it follows that g is an s-wise t-intersecting family. Let
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MAXIMAL INTERSECTING SET FAMILIES
K= G # g G. Let A 1 , ..., A s be arbitrary sets in A. Since g is a generating set, there exist sets G 1 , ..., G s # g, with G i A i for 1 i s. So, we have |K & A 1 & } } } & A s | |K & G 1 & } } } & G s | = |G1 & } } } & G s | t
(by construction of K)
(since g is s-wise t-intersecting).
Therefore, the set K is a kernel of A. Part (3)(b). In Part (1), we showed that g(A) is a generating set of A. By construction, we see that E(A)= G # g(A) G. So, since n 2k&t, by Part (3)(a), it follows that E(A) is a kernel of A. By Part (2), for any K kernel of A, E(A) K, so E(A) is the unique kernel of A with minimum cardinality. K A consequence of this theorem is that we can rewrite n(k, s, t) in terms of essential sets of families in MI s(n, k, t) with n>2k&t. Corollary 2.5. Let k t 1, s 2, and let E max (k, s, t) be given by Emax (k, s, t)=
max
n>2k&t, A # MI s (n, k, t)
|E(A)|.
(4)
Then, n(k, s, t)=Emax (k, s, t). The rest of this section deals with finding a special generating set for families in MI s(n, k, t) that is also a generating set for families in MI s(n$, k, t), for any n$ n. The following is an example of a family A # MI 2(8, 4, 1) for which g(A) is not such a special generating set. For n$ 9, the family U 4n$ (g(A)) is not maximal. Example 2.6. Let g 1 =[[1, 2, 3, 4], [1, 5, 6, 7], [3, 5, 6, 7], [4, 5, 6, 7], [2, 5], [2, 6], [2, 7]]. The set A=U 48 (g 1 ) is in MI 2(8, 4, 1), and E(A)=[1, 7]. The set g(A) as defined in (3) is g(A)=[[1, 2, 3, 4], [1, 5, 6, 7], [3, 5, 6, 7], [4, 5, 6, 7]] _ U 37 ([[2, 5], [2, 6], [2, 7]]) _ U 47 ([[2, 5], [2, 6], [2, 7]]). The set A$=U 49 ( g(A)) is not maximal in I 2(9, 4, 1), since, for example, the set B=[2, 5, 8, 9] intersects every other set in A$, but B Â A$. However, there exists a generating set of A, namely g 1, that has the special property that U 4n (g) # I 2(n, 4, 1) is maximal for all n 8. Now we introduce a new generating set.
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LUCIA MOURA
Proposition 2.7. Let A be a family in MI s(n, k, t), and g(A) be the generating set introduced in (3). Denote by g$(A) the family given by g$(A)=[G G : G # g(A), and [G] _ g(A) is s-wise t-intersecting]. (5) Then, if n>2k&t, the family g$(A) is a generating set for A. Proof. Since g(A) g$(A), we have A=U kn (g(A)) U kn (g$(A)). We have to show U kn (g$(A)) A. Let B # U kn (g$(A)). If A 1 , ..., A s&1 # A, there exist G 1 , ..., G s&1 # g(A), such that A 1 $G 1 , ..., A s&1 $G s&1 . Also, B$G, for G such that [G] _ g(A) is s-wise t-intersecting. So |A 1 & } } } & A s&1 & B| |G 1 & } } } & G s&1 & G| t. Therefore [B] _ A is in I s(n, k, t), and since A is maximal, we have that B # A. K In the next theorem, we show that the generating set g$(B) can be used to produce maximal families for larger n's. Theorem 2.8. Let n n 0 2k&t+1, and let B be a family in MI s(n 0 , k, t). Define A to be a family in I s(n, k, t) given by A=U kn (g$(B)). Then (1)
E(B)=E(A);
(2)
the family A is maximal in I s(n, k, t); and
(3)
g$(B)=g$(A).
Proof. Part (1). Given that B A, we see that E(B) E(A). So, we only need to show that E(A) E(B). Let e # E(A). Then, there exist A1 , ..., A s # A with e # A 1 & } } } & A s and |A 1 & } } } & A s | =t. By construction, Ai $G i for some G i # g$(B), for 1 i s. Using the fact that g$(B) is a generating set for B and Lemma 2.2, we get that |G 1 & } } } & G s | t. Therefore, we must have e # G1 & } } } & G s E(B). Part (2). Clearly A # I s(n, k, t). It is left to show that A is maximal. Let C # [1, n], with |C| =k and such that A _ [C] is s-wise t-intersecting. Considering that g$(B) is a generating set for A, we conclude that g$(B) _ [C] is a generating set for A _ [C]. Thus, by Lemma 2.2, g$(B) _ [C] is s-wise t-intersecting. Given that G # g$(B) G=E(B), we must have that g$(B) _ [C & E(B)] is s-wise t-intersecting. It is easy to prove that C & E(B) # g$(B). Indeed, let B to be any k-extension of C & E(B) contained in [1, n 0 ]. Thus, B _ [B] is s-wise t-intersecting, and since B is maximal, we have B # B. Considering that C & E(B) B & E(B) and that g$(B) _ [C & E(B)] is s-wise t-intersecting we conclude that C & E(B) # g$(B). Therefore, any k-extension of C & E(B) contained in [1, n] belongs to A, by definition; in particular, the set C # A.
MAXIMAL INTERSECTING SET FAMILIES
61
Part (3). First, we show that g$(B) g$(A). Let G # g$(B). Let A be any k-extension of G contained in [1, n]; by definition of A, it follows that A # A. Observing that G E(B)=E(A), we conclude that G A & E(A). Considering that [G] _ g(A) is a generating set for A, by Lemma 2.2, we must have [G] _ g(A) being s-wise t-intersecting. Thus, by the definition of g$(A), we have G # g$(A). Finally, we show that g$(A) g$(B). Let G$ # g$(A). Then, by definition, G$ A & E(A), for some A # A, and [G$] _ g(A) is s-wise t-intersecting. Considering that B A, we see that g(B) g(A). This implies that G$ B & g(B) is s-wise t-intersecting. Now, we just have to show that G$ B & g(B) for some B # B. Let B be any k-extension of A & E(A) contained in [1, n 0 ]. As G$ B, we know that B # A. This implies B is a k-extension of some G # g$(B), by construction of A. In addition, B [1, n 0 ], which implies B # B. We conclude the proof by observing that G$=G$ & E(B) B & E(B). K Now, we show that the generating set g$(A) for a family A is the same as a generating set g$(B) for some family B with smaller n, provided that n is large enough. Theorem 2.9. Let n0 max[2k&t+1, n(k, s, t)], and let A be a family in MI s(n, k, t), with n n 0 . Assume w.l.o.g. that E(A)=[1, |E(A)|]. Let B=U kn0 (g$(A)). Then (1)
E(B)=E(A);
(2)
the family B is maximal in I s(n 0 , k, t); and
(3)
g$(B)=g$(A).
Proof. Part (1). Since B A, it is clear that E(B) E(A). We have to show that E(A) E(B). Let e # E(A), then there exist sets A 1 , ..., A s # A with e # A 1 & } } } & A s and |A 1 & } } } & A s | =t. Let us call I=A 1 & } } } & A s . Consider G i =A i & E(A), for 1 i s. Then G 1 , ..., G s # g(A), and I G 1 & } } } & G s . Thus |G i | |I | =t. Moreover, G 1 , ..., G s # g$(A), so U kn0 ([G 1 , ..., G s ]) B. We choose the following k-extensions of G 1 , ..., G s contained in [1, n 0 ]: v B 1 =G 1 _ L 1 , for some L 1 # [1, n 0 ] "(G 1 _ (G 2 & } } } & G s )); v B i =G i _ L i , for some L i [1, n 0 ] "(G i _ B 1 ), i=2, ..., s. There exists always such L 1 , since |[1, n 0 ] "(G 1 _ (G 2 & } } } & G s ))| n 0 & (|G 1 | + |G 2 & } } } & G s | & |I | ) 2k&t+1&(|G 1 | +k&t) k& |G 1 | . For 2 i s, there exists always such L i , since |[1, n 0 ] "(G i _ B 1 )| =n 0 & (|G i | + |B 1 | & |G i & B 1 | ) 2k&t+1& |G i | &k+t k& |G i | .
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Therefore B 1 , ..., B s # B and B 1 & } } } & B s =(G 1 _ L 1 ) & (G 2 _ L 2 ) & } } } & (G s _ L s ) =(G 1 _ L 1 ) & G 2 & } } } & G s
(by the choice of L 2 , ..., L s )
=(G 1 & G 2 & } } } G s ) _ (L 1 & G 2 & } } } & G s ) =I. So e # B 1 & } } } & B s and |B 1 & } } } & B s | = |I| =t. Therefore e is essential for B. Part (2). One can easily check from B=U kn0 (g$(A)) and G # g$(A) G [1, |E(A)| ] [1, n 0 ], that B # I s(n 0 , k, t). We only have to prove that B is maximal. Let C # [1, n 0 ] such that B$=B _ [C] is s-wise t-intersecting. Then, g = g$(A) _ [C] is a generating set of B$. By Lemma 2.2, the set g is s-wise t-intersecting, which implies that for any G 1 , ..., G s&1 # g$(A), we have |G 1 & } } } & G s&1 & C | t. So, A _ [C] is s-wise t-intersecting, and since A is maximal we must have C # A. Therefore, C & E(A) # g$(A), which implies C # B. Part (3). Let us first prove that g$(A) g$(B). Let G # g$(A). Then, G A & E(A), for some A # A, and g(A) _ [G ] is s-wise t-intersecting. Since B A and E(B)=E(A), we have that g(B) g(A). Thus, g(B) _ [G ] is s-wise t-intersecting. In addition, B=G _ L # B, for any L [1, n 0 ] "G. Then, B & E(B)=(G _ L) & E(A)=G _ (L & E(A)). So, G B & E(B) for some B # B, and g(B) _ [G ] is s-wise t-intersecting. Therefore, G # g$(B). We must show that g$(B) g$(A). Let G # g$(B). Thus, G B & E(B), for some B # B, and g(B) _ [G ] is s-wise t-intersecting. Given that B A and E(B)=E(A), we conclude that G B & E(B)=B & E(A), with B # A. Let G 1 , ..., G s&1 # g(A). Considering that g(A) g$(A) g$(B), we have that G 1 , ..., G s&1 # g$(B). Using the fact that g$(B) is a generating set of B and n 0 >2k&t, by Lemma 2.2, we conclude that g$(B) is s-wise t-intersecting, and in particular that |G & G 1 & } } } & G s&1 | t. Therefore, g(A) _ [G ] is s-wise t-intersecting, which implies G # g$(A). K The following corollary implies that the solution of Problem 1 for k, s, t and n n 0 (k, s, t) can be obtained from the solution of Problem 1 for k, s, t and n=n 0 (k, s, t). Corollary 2.10. Let G(n)=[g$(A) : A # MI s(n, k, t)] and n 0 (k, s, t) =max[n(k, s, t), 2k&t+1]. If n 1 , n 2 n 0 (k, s, t), then G(n 1 ) is isomorphismequivalent to G(n 2 ). Proof. Assume w.l.o.g. that n 2 n 1 . Theorem 2.9 gives that for every family A # MI s(n 2 , k, t), there exist a family B # MI s(n 1 , k, t) such that
MAXIMAL INTERSECTING SET FAMILIES
63
g$(B)t g$(A). Theorem 2.8 gives that for every family B # MI s(n 1 , k, t), there exist a family A # MI s(n 2 , k, t) such that g$(A)= g$(B). K Remark 2.11. For A # MI s(n, k, t), the generating set g$(A) is not necessarily minimal in the sense of set inclusion. For representing A, we may also use the generating set g"(A)=[G # g$(A) : for any G 1 # g$(A), G 1 G implies G 1 =G]. It is easy to check that g"(A) is a generating set of A, and that g"(A) satisfies the same properties as g$(A), given in Theorem 2.9 and Corollary 2.10. Example 2.12.
Let A # MI 2(8, 4, 1) as given in Example 2.6, with
g(A)=[[1, 2, 3, 4], [1, 5, 6, 7], [3, 5, 6, 7], [4, 5, 6, 7]] _ U 37 ([[2, 5], [2, 6], [2, 7]]) _ U 47 ([[2, 5], [2, 6], [2, 7]]). Then, g$(A)= g(A) _ [[2, 5], [2, 6], [2, 7]]], and g"(A)=[[1, 2, 3, 4], [1, 5, 6, 7], [3, 5, 6, 7], [4, 5, 6, 7], [2, 5], [2, 6], [2, 7]].
3. A CONSTRUCTION OF FAMILIES IN MI s(n+1, k+1, t+1) FROM FAMILIES IN MI s((n, k, t) In this section, we show how a family in MI s(n, k, t) can be used to construct a family in MI s(n+1, k+1, t+1), for any s, n, k, and t. We also show that if the construction is applied to two non-isomorphic families, then the resulting families are also non-isomorphic (Theorem 3.5). In addition, for n>2k&t, this construction increases the number of essential numbers by exactly one (Proposition 3.8). The idea behind the construction is as follows. Let A # MI s(n, k, t). The family A(n+1) obtained by adding the element n+1 to every set in A is obviously a family in I s(n+1, k+1, t+1), but not necessarily maximal. The construction adds appropriate sets to A(n+1) in order to obtain a family B that is maximal. In fact, we show that B is the only maximal family in MI s(n+1, k+1, t+1) that contains A(n+1) .
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Construction 3.1. Let A be a family in MI s(n, k, t). Let A(n+1) = [A _ [n+1] : A # A]. For any A # A, denote E(A)=[(C 1 , ..., C s&1 ) # A : |C 1 & } } } & C s&1 & A| =t], and L(A)=( (C1 , ..., Cs&1 ) # E(A) (C 1 & } } } & C s&1 )) "A. Let, for any A # A, A*(A)=
{
[A _ [l ] : l # L(A)] [A _ [i ] : i # [1, n] "A]
if E(A){<, if E(A)=<,
and A*= A # A A*(A). Define B to be the family A(n+1) _ A*. Theorem 3.2. Let A be a family in MI s(n, k, t). Then the family B=A(n+1) _ A* given by Construction 3.1 is a family in MI s(n+1, k+1, t+1). Moreover, family B is the only family in MI s(n+1, k+1, t+1) that contains the subfamily A(n+1) . Proof. First, we will prove thnat B # I s(n+1, k+1, t+1). It is clear that B is (k+1)-uniform, so it remains to prove that B is s-wise (t+1)intersecting. Let B 1 , ..., B s # B, we need to show that |B 1 & } } } & B s | t+1. We split the analysis in two cases. (1)
If B 1 , ..., B s # A(n+1) , then it is trivial that |B 1 & } } } & B s | t+1.
(2) If for some i, 1 i s, B i # A*, we assume w.l.o.g. that B 1 # A*. We know that there exist A 1 , ..., A s # A such that B i $A i , for 1 i s. If |A 1 & } } } & A s | t+1, then we are done. This include the cases when E(A 1 )=<, for in such case |A 1 & C 1 & } } } & C s&1 | t+1, for any C 1 , ..., C s&1 # A. When |A 1 & } } } & A s | =t, we have E(A 1 ){<, and by the construction of A*(A 1 ), we must have B 1 =A 1 _ [l 1 ], with l 1 # L(A 1 ). But, by the construction of L(A 1 ), we must have l 1 # (A 2 & } } } & A s ) "A 1 then |B 1 & } } } & B s | |A 1 & } } } & A s | +1=t+1. We must show that B is a maximal family in I(n+1, k+1, t+1), and B is the only maximal family in I(n+1, k+1, t+1) that contains the subfamily A(n+1) . Let S [1, n+1] be an arbitrary (k+1)-set that is s-wise (t+1)-intersecting with any s&1 sets B 1 , ..., B s&1 # A(n+1) . We prove both of the above statements by showing that S # B. If n+1 # S, then S "[n+1] must be s-wise t-intersecting with any s&1 sets A 1 , ,..., A s&1 # A. Since A is maximal, S "[n+1] # A, and so S # B. If n+1 Â S, then let a # S and call S a =S"[a]. Thus [S] _ A is s-wise (t+1)-intersecting, which implies that [S a ] _ A is s-wise t-intersecting, and then S a # A. Now, if E(S a )=<, by construction of A*(S a ), we have S=S a _ [a] # B. Otherwise, if E(S a ){<, for any (C 1 , ..., C s&1 ) # E(S a ), we have |C 1 & } } } & C s&1 & S a | =t, but |C 1 & } } } & C s&1 & S| t+1, and so we must have a # C 1 & } } } & C s&1 . Therefore a # L(S a ), and then S= Sa _ [a] # B. K
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Example 3.3. Let A # MI 2(6, 3, 1) given by the first column of the table. The set B=A(7) _ A* is in MI 2(7, 4, 2). A
E(A)
[1, 2, 3]
<
[1, 2, 4] [1, 2, 5] [1, 2, 6] [1, 3, 4] [1, 3, 5] [1, 3, 6] [2, 3, 4] [2, 3, 5] [2, 3, 6]
[1, 3, 5] [1, 3, 4] [1, 3, 4] [1, 2, 5] [1, 2, 4] [1, 2, 4] [1, 2, 5] [1, 2, 4] [1, 2, 4]
[1, 3, 6] [1, 3, 6] [1, 3, 5] [1, 2, 6] [1, 2, 6] [1, 2, 5] [1, 2, 6] [1, 2, 6] [1, 2, 5]
[2, 3, 5] [2, 3, 4] [2, 3, 4] [2, 3, 5] [2, 3, 4] [2, 3, 4] [1, 3, 5] [1, 3, 4] [1, 3, 4]
[2, 3, 6] [2, 3, 6] [2, 3, 5] [2, 3, 6] [2, 3, 6] [2, 3, 5] [1, 3, 6] [1, 3, 6] [1, 3, 5]
L(A)
A*(A)
[3] [3] [3] [2] [2] [2] [1] [1] [1]
[1, 2, 3, 4] [1, 2, 3, 6] [1, 2, 3, 4] [1, 2, 3, 5] [1, 2, 3, 6] [1, 2, 3, 4] [1, 2, 3, 5] [1, 2, 3, 6] [1, 2, 3, 4] [1, 2, 3, 5] [1, 2, 3, 6]
A(7) [1, 2, 3, 5]
[1, 2, 3, 7] [1, 2, 4, 7] [1, 2, 5, 7] [1, 2, 6, 7] [1, 3, 4, 7] [1, 3, 5, 7] [1, 3, 6, 7] [2, 3, 4, 7] [2, 3, 5, 7] [2, 3, 6, 7]
Now we introduce some notation that will be used in the next theorem, as well as in the sections to follow. Notation 3.4. For any family B of subsets of [1, n], and for any x, y, z # [1, n], we denote Bx =[B # B : x # B], Ba, b =Ba & Bb ,
for
a, b # [x, x, y, y ],
Bx =B "Bx , Ba, b, c =Ba & Bb & Bc ,
for
a, b, c # [x, x, y, y, z, z ].
The following theorem shows that if Construction 3.1 is applied to nonisomorphic families, then the resulting families are also non-isomorphic. Theorem 3.5. Let A 1, A 2 # MI s(n, k, t), and let B 1 and B 2 be the families in MI s(n+1, k+1, t+1) given by the application of Construction 3.1 to the families A 1 and A 2, respectively. Then B 1 tB 2 implies A 1 tA 2. Proof. Let ? be the permutation on [1, n+1] such that B 2 =?(B 1 ). If ?(n+1)=n+1, then trivially A 2 =?(A 1 ). Otherwise, we can assume w.l.o.g. that ?(n+1)=n. Let us call B=B 2 =?(B 1 ). For a family C we denote C "[x]=[C "[x] : C # C]; we can then write ?(A 1 )=Bn, n+1 "[n] _ Bn, n+1 "[n] A 2 =Bn, n+1 "[n+1] _ Bn, n+1 "[n+1]. We want to show ?(A 1 )tA 2. It is clear that Bn, n+1 "[n]t Bn, n+1 "[n+1). It is enough to prove that Bn, n+1 "[n]=Bn, n+1 "[n+1].
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We claim that for any A # Bn, n+1 "[n], the set [A ] _ A 2 is s-wise t-intersecting. By definition, A 2 is s-wise t-intersecting. Now, let A 1 , ..., A s&1 # A 2 and let B i =A i _ [n+1], for 1 i s&1, B =A _ [n]. Observe that B 1 , ..., B s&1 , B # B. Let us call I A =A 1 & } } } & A s&1 & A and I B =B 1 & } } } & B s&1 & B. First, we observe that I A =I B"[n, n+1], and since B # Bn, n+1 , we known that n+1 Â I B . So |I A | = |I B"[n]| (t+1)&1=t. So, [A ] _ A 2 is s-wise t-intersecting. Then, since A 2 is maximal it follows that A # A 2. Therefore, Bn, n+1"[n] A 2, which means Bn, n+1"[n] Bn, n+1"[n+1]. By an analogous argument, we show that Bn, n+1"[n+1] Bn, n+1"[n]. Therefore, Bn, n+1"[n+1]=Bn, n+1"[n], which completes the proof. K An immediate consequence of the previous theorem is that, for any s 2, the number of non-isomorphic families is non-decreasing, when parameters n, k and t are increased by one. Recall the meaning of D(}) from Notation 1.2. Corollary 3.6. For any s, n, k and t 1, we have D(MI s(n+1, k+1, t+1)) D(MI s(n, k, t)). The following lemma is a well-known result for s=2 (see, e.g., [6]), saying basically that, for n 2k&t, if A # MI s(n, k, t), then A Â MI s(n, k, t+1). Lemma 3.7.
Let n 2k&t and A # MI s(n, k, t). Then E(A){<.
Now we show that when we apply the previous construction for n 2k&t, then the number of essential numbers increases by exactly one. Proposition 3.8. Let n 2k&t. Let A be a family in MI s(n, k, t), and let the family B # MI s(n+1, k+1, t+1) be the family given by Construction 3.1. Then E(B)=E(A) _ [n+1]. Sketch of the Proof. Lemma 3.7 is used to prove the inclusion E(A) _ [n+1] E(B). The inclusion E(B) E(A) _ [n+1] is proven by showing that for any essential s-tuple for e in B, e{n+1, one can carefully remove one element from each set of the s-tuple producing an essential s-tuple for e in A. K 4. ALL MAXIMAL k-UNIFORM PAIRWISE t-INTERSECTING FAMILIES FOR k t+2 In this section, we solve Problem 1 for the case of s=2 and k=t+1, t+2 (see Fig. 1). The case of k=t+1 is simple, but we include it here for the sake of completeness.
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67
FIG. 1. Determination of all maximal families for s=2, k=t+2, for all t 1 (see Theorem 4.8).
4.1. Determination of All Families in MI 2(n, t+1, t) In this subsection, we determine all families A # MI 2(n, t+1, t) for arbitrary n t+2, and show that n(t+1, 2, t)=t+2, for all t 1. We will use the following notation, for any k and n: ( [n] k )= [B [1, n] : |B| =k]. Proposition 4.1. following hold: (1)
Let t 1 and n t+2, and A # MI 2(n, t+1, t). The
if n=t+2, then A=( [t+2] t+1 );
[n] ) : B$[1, t]] or (2) if n t+3, then either AtB$=[B ( t+1 ). At( [t+2] t+1
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Sketch of the Proof. Part (1). The set ( [t+2] t+1 ) containing every possible subset is t-intersecting. Part (2). The proof consists of a simple case analysis. In the case that there exist distinct sets A 1 , A 2 , A 3 # A with |A 1 & A 2 & A 3 | =t it is easy to show that AtB$. In the complementary case, we conclude that sets in A must be contained in a fixed (t+2)-set. K Thus, by Part (1), we have At( [t+2] t+1 ). Corollary 4.2. For t 1, we have n(t+1, 2, t)=t+2. Proof. From Corollary 2.5, we have n(t+1, 2, t)=E max (t+1, 2, t), but by Proposition 4.1 we have E max (t+1, 2, 5)=t+2. K 4.2. Determination of All Families in MI 2(n, t+2, t) In this subsection, we show that Construction 3.1 is enough to generate all the sets in MI 2(n, t+3, t+1) when applied to all the sets in MI 2(n, t+2, t), for any n and t 2 (Theorem 4.5). We also show that n(t+2, 2, t)=t+6 (Theorem 4.7). Using these two results combined with results from previous sections, we manage to reduce Problem 1 (for any s=2, n, t and k=t+2) to the solution of Problem 1 for (s, n, k, t) # [(2, 7, 3, 1), (2, 8, 4, 2)] (Theorem 4.8). This solution is obtained by computer and shown in Table I. Throughout this section, we will largely use Notation 3.4. In addition, for any A # I s(n, k, t), and a # [1, n], we denote by n a the number of sets in A containing a, that is n a = |Aa | . Lemma 4.3. Let A be a family in MI 2(n, t+2, t), for t 3. Let x # [1, n] such that n x =max i # [1, n] n i . Then the family Ax is pairwise (t+1)-intersecting. Proof. It is clear that Ax is pairwise t-intersecting, since Ax A. In order to prove that Ax is, in addition, pairwise (t+1)-intersecting, we assume by contradiction that there exist A 1 , A 2 # Ax , such that |A 1 & A 2 | =t. Then A 1 and A 2 must be of the form A 1 =[ y 1 , y 2 , ..., y t , : 1 , ; 1 ],
and
A 2 =[ y 1 , y 2 , ..., y t , : 2 , ; 2 ],
for distinct y 1 , y 2 , ..., y t , : 1 , ; 1 , : 2 , ; 2 , different from x, For any i, since n yi n x we must have |Ayi , x | |Ayi , x | 2.
(6)
We claim that Ayi , x cannot be pairwise (t+1)-intersecting. Indeed, if it was so, it would imply that the family obtained by replacing x by y i in Ayi , x , denoted by Ayi , x &[x]+[ y i ], would t-intersect all other sets in A.
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Then, since A is maximal, we would have Ayi , x &[x]+[y i ] A. Consequently, we would get |Ayi , x | |(Ayi , x &[x]+[ y i ]) _ [A 1 ] _ [A 2 ]| = |Ayi , x | +2> |Ayi , x | , contradicting (6). Therefore, for i=1, ..., t, there exist B i1 , B i2 # Ayi , x , with |B i1 & B i2 | =t.
(7)
Since B i1 , B i2 # Ayi , x and must also t-intersect both A 1 and A 2 , they must be of the form B i1 =[x, y 1 , y 2 , ..., y t ]"[ y i ] _ Z i1 ,
B i2 =[x, y 1 , y 2 , ..., y t ]"[ y i ] _ Z i2 ,
with Z i1 , Z i2 # [[: 1 , : 2 ], [: 1 , ; 2 ], [; 1 , : 2 ], [; 1 , ; 2 ]]. Now, condition (7) translates into Z i1 & Z i2 =<, for all i=1, ..., t, which implies [Z i1 , Z i2 ] equals either [[: 1 , : 2 ], [; 1 , ; 2 ]] or [[: 1 , ; 2 ], [; 1 , : 2 ]]. Since t 3, by i i i i the pigeon-hole principle for some i 1 {i 2 , we have [Z 11, Z 21 ]=[Z 12, Z 22 ]. i1 i2 i1 i2 So, w.l.o.g., Z 1 =Z 1 , which implies |B 1 & B 2 | =t&1
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B & L${<. Indeed, let A$ # A$. Then |A$| =t+1, |A$ & B| t 3 and |A$ & L$| t&1. Then, |B & L$| |A$ & B & L$| |A$ & B| + |A$ & L$| & |A$| t+(t&1)&(t+1)=t&2 1. Let y # B & L$. By Lemma 4.4, (B"[ y] _ [x]) # A and so (B"[ y]) # A$. By definition of L$, we have |(B "[ y]) & L$| t&1. Since y # B & L$, we get |B & L$| t, and finally |B & L| t. K We examine the consequences of the previous theorem on the size of kernels for s=2 and t=k&2 2. Corollary 4.6. Let t=k&2 2. Then n(t+2, 2, t)=n(4, 2, 2)+t&2. Proof. Let n t+5 and let B # MI 2(n, t+2, t). By Theorem 4.5, there exists A # MI 2(n&(t&2), 4, 2) such that each B is obtained by t&2 applications of Construction 3.1 to A. Thus, by Proposition 3.8, we have |E(B)| = |E(A)| +t&2. Finally, using Corollary 2.5, we get n(t+2, 2, t)= n(4, 2, 2)+t&2. K Finally, we just need to determine n(3, 2, 1) and n(4, 2, 2) in order to obtain n(t+2, 2, t) for any t 1. Theorem 4.7.
For any t 1, we have n(t+2, 2, t)=t+6.
Proof. First, we observe that n(3, 2, 1) 7 and n(4, 2, 2) 8, since families G 8 # MI 2(7, 3, 1) and H 8 # MI 2(8, 4, 2) (see Table I) are such that E(G 8 )=7 and E(H 8 )=8. In addition, if n(4, 2, 2) 8, using that n(k&1, s, t&1) n(k, s, t)&1, we get that n(3, 2, 1) 7. This implies n(3, 2, 1)=7 and n(4, 2, 2) = 8, which combined with Corollary 4.6, leads to n(t+2, 2, t)=t+6, for t 1. Thus, it is enough to show that n(4, 2, 2) 8. Let A # MI 2(n, 4, 2) for arbitrary n 8, and let A, B # A be an essential pair for A (there exists such a pair by Lemma 3.7). We can assume w.l.o.g. that A=[1, 2, 3, 4] and B=[1, 2, 5, 6]. We can also assume that A has no set of the form [1, 2, a, b] with a, b # [1, n] " [1, 6], for this would imply that [1, 2]. We can also assume that A has no set of the form [1, 2, a, b] with a, b # [1, n] " [1, 6], for this would imply that [1, 2] must be contained in every other set in A and then |E(A)| =2. Therefore, the remaining sets that are possibly contained in A and that include an element x  [1, 6] are of the form [1, 2, a, x],
for
a # [3, 4, 5, 6]
[b, c, e, x],
for
b # [1, 2], [c, e] [[3, 5], [3, 6], [4, 5], [4, 6]], and x # [7, n].
and
x # [7, n];
(8)
(9)
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Therefore, any essential element x # [7, n] must have an essential pair chosen from sets in (8) and (9). We will show there are at most 2 such essential elements, which implies |E(A)| 8. Define a graph G which vertices corresponding to the possible triples accompanying an essential element x # [7, n], given by (8) and (9), and edges placed whenever the triples corresponding to its endings intersect. Colour the edges whose triples corresponding to its endings intersect in exactly 1 element. Now, for each essential element x # [7, n] for A, let e(x) be an edge of G connecting some triples T1 and T2 such that T1 _ [x] and T2 _ [x] is an essential pair for A. Let E$=[e(x) : x # E(A) " [1, 6]]. Let G$ be the subgraph of G induced by the set of all vertices that are endings for some edge in E$. Since A is 2-intersecting, G$ is a complete graph whose coloured edges form a matching. We will show that such E$ must have |E$| 2, which implies |E(A)| 8. We reduce the size of the subgraphs we have to look at, by observing that another essential pair for x # [7, n], must be equivalent to one of the following cases: C 1 =[1, 3, 5, x ], D 1 =[1, 2, 4, x ]; C 2 =[1, 3, 5, x ], D 2 =[1, 4, 6, x ]; or C 3 =[1, 3, 5, x ], D 3 =[2, 3, 6, x ]. For i=1, 2, 3, we analyse the case of e(x ) joining v Ci and v Di , the vertices corresponding to C i and D i , respectively. Let Gi be the subgraph of G that is induced by v Ci , v Di and every vertex v that have non-coloured edges incident to both v Ci and v Di . The edges in E$ are coloured and are contained in Gi for some 1 i 3. In Fig. 2, we show graphs G1 , G2 and G3 with thick edges representing the coloured edges. By inspection, we conclude that |E$ | 2. K The following theorem is a combination of many of the previous results. Theorem 4.8. The non-isomorphic families in MI 2(8, 4, 2) and in MI 2 (7, 3, 1) completely determine all non-isomorphic families in MI 2(n, t+2, t), for all n t+6 and t 1.
FIG. 2.
Coloured edges of graphs G1 , G2 , and G3 .
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Proof. Suppose we have listed all non-isomorphic families in MI 2(8, 4, 2) and MI 2(7, 3, 1). We outline how to obtain all nonisomorphic sets in MI 2(n, t+2, t) for arbitrary n t+6, t 1. If t=1, by our assumptions, we have listed all non-isomorphic families in MI 2(t+6, t+2, t)=MI 2(7, 3, 1). If t 2, perform t&2 applications of Construction 3.1 to each of the non-isomorphic sets in MI 2(8, 4, 2). By Theorem 3.5 and Theorem 4.5, this gives all non-isomorphic sets in MI 2(8+(t&2), 4+(t&2), 2+(t&2))=MI 2(t+6, t+2, t). Now, by Theorem 4.7, we have that n 0 (t+2, t)=t+6. Corollary 2.10 implies that computing generating sets g$(A), as defined in (5), for all non-isomorphic sets in MI 2(t+6, t+2, t) gives generating sets for all non-isomorphic sets in MI 2(n, t+2, t), for n t+6. K Corollary 4.9. Let t 1, and n t+6. Then, D(MI 2(n, t+2, t))=
{
D(MI 2(7, 3, 1))=15, D(MI 2(8, 4, 2))=17,
if t=1, if t 2.
A complete list of non-isomorphic sets in MI 2(8, 4, 2) and in MI 2(7, 3, 1) is given in Table I.
ACKNOWLEDGMENTS This article is part of my Ph.D. research at University of Toronto under the supervision of Professor Rudi Mathon. I thank my supervisor for the helpful discussions during the preparation of this manuscript. I also thank an anonymous referee for valuable suggestions.
REFERENCES 1. R. Ahlswede and L. H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996), 121138. 2. R. Ahlswede and L. H. Khachatrian, The complete intersection theorem for systems of finite sets, European J. Combin. 18 (1997), 127136. 3. N. Alon and Z. Furedi, On the kernel of intersecting families, Graphs Combin. 3 (1987), 9194. 4. M. Ca*czynska-Kar*owicz, Theorem on families of finite sets, Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys. 12 (1964), 8789. 5. A. Ehrenfeucht and J. Mycielski, Interpolation of functions over a measure space and conjectures about memory, J. Approx. Theory 9 (1973), 218236. 6. P. Erdo s, C. Ko, and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford 12 (1961), 313320. 7. P. Erdo s and L. Lovasz, Problems and results on 3-chromatic hypergraphs and some related questions, in ``Infinite and Finite Sets'' (A. Hajnal, R. Rado, and V. T. Sos, Eds.), Proc. Colloq. Math. Soc. J. Bolyai, Vol. 10, pp. 609627, Keszthely, Hungary, 1973.
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8. P. Frankl, On intersecting families of finite sets, J. Combin. Theory Ser. A 24 (1978), 146161. 9. P. Frankl, Multiple-intersecting families, J. Combin. Theory Ser. B 53 (1991), 195234. 10. P. Frankl and Z. Furedi, Finite projective spaces and intersecting hypergraphs, Combinatorica 6 (1986), 335354. 11. Z. Furedi, Geometrical solution of an intersection problem for two hypergraphs, European J. Combin. 5 (1984), 133136. 12. A. J. W. Hilton and E. C. Milner, Some intersection theorems for systems of finite sets, Quart. J. Math. Oxford 18 (1967), 369384. 13. J. Kahn and P. Seymour, unpublished. 14. Y. Kohayakawa, A note on kernels of intersecting families, European J. Combin. 11 (1990), 155164. 15. L. Moura, Polyhedral methods in design theory, in ``Computational and Constructive Design Theory'' (W. D. Wallis, Ed.), pp. 227254, Kluwer Academic, Dordrecht, 1996. 16. Zs. Tuza, Critical hypergraphs and intersecting set-pair systems, J. Combin. Theory Ser. B 39 (1985), 134145.
Maximal s-Wise t-Intersecting Families of Sets: Kernels, Generating Sets, and Enumeration Lucia Moura Department of Computer Science, University of Toronto, Toronto, Canada, M5S 3G4 E-mail: luciacs. toronto.edu Communicated by the Managing Editors Received July 28, 1997
A family A of k-subsets of an n-set is said to be s-wise t-intersecting if |A 1 & } } } & A s | t, for any A 1 , ..., A s # A. For fixed s, n, k, and t, let I s(n, k, t) denote the set of all such families. A family A # I s(n, k, t) is said to be maximal if it is not properly contained in any other family in I s(n, k, t). We show that for fixed s, k, t, there is an integer n0 =n 0 (k, s, t), for which the maximal families in I s(n 0 , k, t) completely determine the maximal families in I s(n, k, t), for all n n 0 . We give a construction for maximal families in I s(n+1, k+1, t+1) based on those in I s(n, k, t). Finally, for s=2, we classify the maximal families for k=t+1, n t+2, t 1, and for k=t+2, n t+6, t 1. The concepts of kernels and generating sets of a family of subsets play an important role in this work. 1999 Academic Press, Inc.
1. MOTIVATION, DEFINITIONS, AND RESULTS For i, j # N, we abbreviate the set [i, i+1, ..., j] as [i, j]. Given s 2 and n, t 1, a family A of subsets of [1, n] is said to be s-wise t-intersecting, if any s members A 1 , ..., A s of A are such that |A 1 & } } } & A s | t. A family A is said to be k-uniform if every member of A has cardinality k. We denote by I s(n, k, t) the set of all k-uniform s-wise t-intersecting families of subsets of [1, n]. We are interested in the families in I s(n, k, t) that are maximal with respect to set inclusion, i.e., the families A # I s(n, k, t) such that for any B # I s(n, k, t), if B$A then B=A. Throughout this paper, we denote by MI s(n, k, t) the set of all families in I s(n, k, t) that are maximal in this sense. The problem that motivated this paper is the following. Problem 1. Given s, n, k, and t, list all families of MI s(n, k, t); or alternatively, list all non isomorphic families of MI s(n, k, t). 52 0097-316599 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved.
MAXIMAL INTERSECTING SET FAMILIES
53
We should make clear what we mean by isomorphic families. Definition 1.1. We say that two families A 1 and A 2 of subsets of [1, n] are isomorphic, denoted by A 1 tA 2, if there exist a permutation ? of the set [1, n], such that ? sends A 1 to A 2. We also write A 2 =?(A 1 ). Problem 1 is relevant for the study of some polyhedra arising in combinatorial design theory. The families in MI 2(n, k, t) give rise to facets of the polyhedron for the t-(n, k, 1) packing designs, and are also related to the polyhedron for the t-(n, k, 1) designs, also called S(t, k, n) Steiner systems. Precise definitions of these structures as well as of their corresponding polyhedra can be found in [15]. Problem 1 is also of independent interest in extremal set theory. Related works dealing with the determination of the largest family in MI s(n, k, t) include [1, 2, 6, 9, 12]. We make use of two important concepts in the study of families in MI s(n, k, t), namely kernels and generating sets. First, we define the notion of a generating set of a family of subsets, as introduced by Ahlswede and Khachatrian [1, 2]. Let us fix some notation. For any B [1, n], let U kn (B)=[C [1, n] : C$B, |C| =k], and for a family B of subsets of [1, n], let U kn (B)= B # B U kn (B). Let A be a k-uniform family of subsets of [1, n]; a family g of subsets of [1, n] is called a generating set of A, if U kn ( g)=A. The set A itself is a generating set of A, although not a very interesting one. We would like to be able to find generating sets g that are ``minimal'' in some sense, for example, the ones for which | G # g G | or | g | are minimal. Such generating sets provide us with a compact way to represent families of subsets, which proved to be useful for studying Problem 1. Now, we turn our attention to the concept of a kernel of an s-wise t-intersecting family of subsets. For a family A in I s(n, k, t), a set K [1, n] is a kernel of A if any s sets in A meet in at least t elements of K, i.e., for any A 1 , ..., A s # A we have |A 1 & } } } & A s & K| t. Denote by n(k, s, t) the smallest integer for which every family A in n I s(n, k, t) has a kernel K(A) with |K(A)| n(k, s, t) (the standard definition of n(k, s, t) does not require the families A to be k-uniform, but rather having members with cardinality at most k; both definitions turn out to be equivalent). It is not immediate to conclude that n(k, s, t) is finite for all k, s and t. Ca*czynska-Kar*owicz [4] proved that n(k, 2, 1) is finite. The first explicit upper bound on n(k, 2, 1) was given by Ehrenfeucht and Mycielski [5]. Lower bounds and further improvements on upper bound were given by Erdo s and Lovasz [7] and Tuza [16]. The finiteness of n(k, s, t) was independently proven by Frankl [8], in an implicit form, Kahn and Seymour [13], and Frankl and Furedi [10]. These results imply that at least t elements of each of the s-wise intersections of a family A # I s(n, k, t) are included in a small subset (with cardinality independent of n) of the underlying
54
LUCIA MOURA
set [1, n]. General bounds on n(k, s, t) are given by Alon and Furedi [3], who proved that 1 13 (k&t)(s&1) k&t+1+w(k&t)(s&1)x n(k, s, t) k (3 ) 3 w(k&t)(s&1)x k
\
\ +
ss (s&1) s&1
+
w(k&t)(s&1)x
.
(1)
They give an explicit construction for the lower bound, and also show that the upper bound follows from a theorem of Furedi [11]. Kohayakawa [14] sharpened the lower bound in (1). We note that the upper bound in (1) can be improved by a simple observation on the last step of Alon and Furedi's proof. Indeed, in that proof [3, Theorem 2.2], they obtain an upper bound on the maximum size of a kernel given by the size of the union of m sets of size k, with m=
\
k&t+1+w(k&t)(s&1)x . w(k&t)(s&1)x
+
This leads to the upper bound in (1). However, the m sets form an s-wise t-intersecting family, which implies that every set has at least t elements in common with a fixed set, say the first one. Therefore, the upper bound can be improved to k&t+1+w(k&t)(s&1)x +t w(k&t)(s&1)x
\ s (k&t) \(s&1) +
n(k, s, t) (k&t)
s
s&1
+
w(k&t)(s&1)x
+t.
(2)
Before summarizing the contributions in this paper, we fix some notation and give some definition. Notation 1.2. Let C be a collection of families of subsets. We denote by D(C) the number of non isomorphic members of C, that is, the number of equivalent classes induced by the equivalent relation given by t. Definition 1.3. Let C 1, C 2 be collections of families of subsets. We say that collections C 1 and C 2 are isomorphism-equivalent, if for every family A 1 # C 1 there exists a family A 2 # C 2 such that A 2 tA 1, and vice versa.
55
MAXIMAL INTERSECTING SET FAMILIES
In Section 2, we study the relationship between kernels and generating sets of families in MI s(n, k, t). We introduce the notion of a set of essential elements E(A) [1, n] for a family A # I s(n, k, t). We show that for maximal families the concept of kernels, generating sets, and essential elements are closely related (Theorem 2.4). In particular, for n>2k&t and A # MI s(n, k, t), we exhibit generating sets g(A) and g$(A) of A such that E(A)= G # g(A) G= G # g$(A) G is the unique kernel of A with minimum cardinality. We also show that for n large enough, more precisely, for n n 0 (k, s, t)=max[2k&t+1, n(k, s, t)], the set formed by the generating sets g$(A) for all A # MI s(n, k, t) is isomorphism-equivalent to the set formed by the generating sets g$(B) for all B # MI s(n 0 (k, s, t), k, t). Therefore, TABLE I All Generating Sets for s=2 and k=t+2 Generating sets for all non-isomorphic families in MI 2(n, 3, 1), for n 7 Gi
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
[1] [1, 2] [1, 2] [1, 2] [1, 2] [1, 2] [1, 2] [1, 2, 3] [1, 2, 3] [1, 2, 3] [1, 2, 3] [1, 2, 3] [1, 2, 4] [1, 2, 3] [1, 2, 3]
[1, 3] [1, 3] [1, 3] [1, 3, 4] [1, 3, 4] [1, 3, 4] [1, 4, 5] [1, 2, 4] [1, 2, 4] [1, 2, 4] [1, 2, 4] [1, 2, 5] [1, 2, 4] [1, 2, 4]
[2, 3] [1, 4] [1, 4, 5] [1, 3, 5] [1, 3, 6] [1, 3, 6] [1, 6, 7] [1, 2, 5] [1, 2, 5] [1, 2, 5] [1, 2, 5] [1, 2, 6] [1, 3, 6] [1, 2, 5]
[2, 3, 4] [2, 3, 4] [1, 4, 5] [1, 4, 5] [1, 4, 5] [2, 4, 6] [1, 3, 4] [1, 3, 4] [1, 3, 4] [1, 3, 5] [1, 3, 4] [1, 4, 5] [1, 3, 4]
[2, 3, 5] [2, 3, 4] [1, 5, 6] [2, 3, 4] [2, 5, 7] [1, 3, 5] [1, 3, 6] [1, 3, 6] [1, 3, 6] [1, 3, 5] [1, 5, 6] [1, 3, 6]
[2, 3, 5] [2, 3, 5] [2, 3, 5] [3, 4, 7] [1, 4, 5] [1, 4, 5] [1, 4, 5] [1, 4, 6] [1, 3, 6] [2, 3, 5] [1, 4, 7]
[2, 4, 5] [2, 4, 6] [2, 4, 6] [3, 5, 6] [2, 3, 4] [1, 5, 6] [2, 3, 4] [2, 3, 4] [2, 3, 4] [2, 4, 6] [2, 3, 4]
[2, 3, 5] [2, 3, 5] [2, 3, 5] [2, 3, 6] [2, 3, 5] [2, 5, 6] [2, 3, 7]
[2, 4, 5] [2, 4, 6] [2, 4, 6] [2, 5, 6] [2, 3, 6] [3, 4, 5] [2, 4, 6]
[3, 4, 5] [3, 4, 5] [3, 4, 5] [3, 4, 5] [4, 5, 6] [3, 4, 6] [3, 4, 5]
Generating sets for all non-isomorphic families in MI 2(n, 4, 2), for n 8 Hi
i
One application of Construction 3.1 to G i
1 ... 15 16 17
[1, 2, 7] [1, 2, 3, 7]
[1, 3, 4, 7] [1, 4, 5, 7]
[1, 5, 6, 7] [2, 4, 6, 7]
[2, 3, 6, 7] [3, 5, 6, 7]
[2, 4, 5, 7] [1, 2, 5, 6]
[1, 2, 3, 5] [1, 3, 4, 6]
[1, 2, 4, 6] [2, 3, 4, 5]
Generating sets for all non-isomorphic families in MI 2(n, t+2, t), for t 3, n t+6 i 1 ... 17
Li (t&2) applications of Construction 3.1 to H i
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LUCIA MOURA
given k, s and t, in order to solve Problem 1 for n n 0 (k, s, t), it is enough to list the generating sets g$ for the solutions obtained for n=n 0 (k, s, t). In Section 3, we give a general construction for families in MI s(n+1, k+1, t+1) based on families in MI s(n, k, t). This construction produces non-isomorphic families when applied to non-isomorphic ones. This implies that D(MI s(n+l, k+l, t+l)) D(MI s(n, k, t)), for all l 0. In Section 4, we solve Problem 1 for s=2 and k t+2. The case of k=t+1 is simple, but is included for the sake of completeness. The solution of Problem 1 for k=t+2 is more interesting. We show that the construction given in Section 3, when applied to all sets in MI 2(n, t+2, t), gives all the sets in MI 2(n+1, t+3, t+1), for all n and t 2. This reduces the case k=t+2 to the determination of families in MI 2(n 0 (4, 2, 2), 4, 2) and MI 2(n 0 (3, 2, 1), 3, 1), which were obtained by computer. The upper bound in (2) gives n(t+1, 2, t) t+3 and n(t+2, 2, t) t+20, but we prove that n(t+1, 2, t)=t+2 and n(t+2, 2, t)=t+6, for any t 1. Moreover, the results regarding enumeration of non-isomorphic sets for the case of k&t 2 are as follows. For any t 1 we have D(MI 2(n, t+1, t))=
2,
{1,
for n t+2, for n
The form of such sets are given in Theorem 4.1. For any n t+6, we have D(MI 2(n, t+2, t))=
15,
{17,
for t=1, for t 2.
The generating sets for the corresponding families are given in Table I.
2. KERNELS AND GENERATING SETS We begin this section by introducing the new concept of a number in [1, n] being essential for a family in I s(n, k, t). Then, in Theorem 2.4, we show some properties relating kernels, generating sets, and essential numbers for families in MI s(n, k, t). In Corollary 2.5, we rewrite n(k, s, t) based on essential numbers for families in MI s(n, k, t) with n>2k&t. Then, we exhibit some special generating sets (given in (5)) for families in MI s(n, k, t) with n n 0 (k, s, t)=max[2k&t+1, n(k, s, t)], that allows us to show that a solution for Problem 1 for k, s, t and n n 0 (k, s, t) can be obtained from the solution for Problem 1 for k, s, t and n=n 0 (k, s, t) (Corollary 2.10).
MAXIMAL INTERSECTING SET FAMILIES
57
Definition 2.1. Let A # I s(n, k, t). We say that e # [1, n] is essential for A, if there exist sets A 1 , ..., A s with e # A 1 & } } } & A s such that |A 1 & } } } & A s | =t. The s-tuple (A 1 , ..., A s ) is called an essential s-tuple for e in A. We also say that e is essential for A j relatively to A, for any 1 j s. The essential set of A is defined by E(A)=[e # [1, n] : e is essential for A]. It is clear from the above definition that every kernel of a family A # I s(n, k, t) must contain E(A). In Theorem 2.4, we show that for a family A # MI s(n, k, t) with n>2k&t, the set E(A) is the unique kernel of A with minimum cardinality. We also exhibit a generating set g(A) of A that is formed by elements in E(A), and that is largely used in the rest of the paper. For proving Theorem 2.4, we need the following two lemmas. The first one generalizes a known result for s=2 (see [2]). Lemma 2.2. Let n>2k&t. If A is a family in I s(n, k, t) and g is a generating set of A, then g is an s-wise t-intersecting family. Proof. Suppose by contradiction that g is not s-wise t-intersecting. Then there exist sets G 1 , ..., G s # g such that |G 1 & } } } & G s |
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LUCIA MOURA
Proof. We will show that B # A by induction on |B"A|. If |B"A| =0, trivially B=A # A. If |B "A| 1, then choose x # B" A and y # A "B. Let B$=(B" [x]) _ [ y]. Since B$$SA and |B$"A| <|B" A| , by induction hypothesis B$ # A. Since y is not essential, B$" [ y] is t-intersecting with every (s&1)-tuple of sets in A, and so is B. Therefore, since A is maximal, it follows that B # A. K The following theorem establishes connections between kernels, generating sets and essential numbers for a family A in MI s(n, k, t). In particular, we introduce a generating set formed by essential elements of A, and we show that when n>2k&t the essential set of A is the unique kernel of A with minimum cardinality. We observe that the assumption n>2k&t is not so restrictive. For example, when s=2, if n 2k&t, then the family formed by every k-subset of [1, n] is pairwise t-intersecting, and therefore is the only family in MI 2(n, k, t). Theorem 2.4. Let A be a family in MI s(n, k, t). (1)
Let us denote by g(A), the family given by g(A)=[A & E(A) : A # A].
(3)
Then, g(A) is a generating set for A. (2)
For any kernel K of A we have that E(A) K.
(3)
If n>2k&t, then:
(a)
for any generating set g of A, the set G # g G is a kernel of A;
(b) the set E(A) is a kernel of A; moreover, E(A) is the unique kernel of A with minimum cardinality. Proof. Part (1). Trivially A U kn (g(A)), so it remains to prove that U (g(A)) A. Take B # U kn (g(A)). Then, B=S _ L with S # g(A) and L [1, n] "S. Since S # g(A), then there exists an A # A with S=A & E(A). Therefore, since B$S=A & E(A), by Lemma 2.3, B # A. k n
Part (2). Let e # E(A) and K any kernel of A. Since e is essential for A there exist sets A 1 , ..., A s # A such that e # A 1 & } } } & A s and |A 1 & } } } A s | =t. In addition, since K is a kernel of A, |K & A1 & } } } & A s | t and so e # K. Part (3)(a). Let g be a generating set of A. Since n 2k&t, by Lemma 2.2, it follows that g is an s-wise t-intersecting family. Let
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MAXIMAL INTERSECTING SET FAMILIES
K= G # g G. Let A 1 , ..., A s be arbitrary sets in A. Since g is a generating set, there exist sets G 1 , ..., G s # g, with G i A i for 1 i s. So, we have |K & A 1 & } } } & A s | |K & G 1 & } } } & G s | = |G1 & } } } & G s | t
(by construction of K)
(since g is s-wise t-intersecting).
Therefore, the set K is a kernel of A. Part (3)(b). In Part (1), we showed that g(A) is a generating set of A. By construction, we see that E(A)= G # g(A) G. So, since n 2k&t, by Part (3)(a), it follows that E(A) is a kernel of A. By Part (2), for any K kernel of A, E(A) K, so E(A) is the unique kernel of A with minimum cardinality. K A consequence of this theorem is that we can rewrite n(k, s, t) in terms of essential sets of families in MI s(n, k, t) with n>2k&t. Corollary 2.5. Let k t 1, s 2, and let E max (k, s, t) be given by Emax (k, s, t)=
max
n>2k&t, A # MI s (n, k, t)
|E(A)|.
(4)
Then, n(k, s, t)=Emax (k, s, t). The rest of this section deals with finding a special generating set for families in MI s(n, k, t) that is also a generating set for families in MI s(n$, k, t), for any n$ n. The following is an example of a family A # MI 2(8, 4, 1) for which g(A) is not such a special generating set. For n$ 9, the family U 4n$ (g(A)) is not maximal. Example 2.6. Let g 1 =[[1, 2, 3, 4], [1, 5, 6, 7], [3, 5, 6, 7], [4, 5, 6, 7], [2, 5], [2, 6], [2, 7]]. The set A=U 48 (g 1 ) is in MI 2(8, 4, 1), and E(A)=[1, 7]. The set g(A) as defined in (3) is g(A)=[[1, 2, 3, 4], [1, 5, 6, 7], [3, 5, 6, 7], [4, 5, 6, 7]] _ U 37 ([[2, 5], [2, 6], [2, 7]]) _ U 47 ([[2, 5], [2, 6], [2, 7]]). The set A$=U 49 ( g(A)) is not maximal in I 2(9, 4, 1), since, for example, the set B=[2, 5, 8, 9] intersects every other set in A$, but B Â A$. However, there exists a generating set of A, namely g 1, that has the special property that U 4n (g) # I 2(n, 4, 1) is maximal for all n 8. Now we introduce a new generating set.
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LUCIA MOURA
Proposition 2.7. Let A be a family in MI s(n, k, t), and g(A) be the generating set introduced in (3). Denote by g$(A) the family given by g$(A)=[G G : G # g(A), and [G] _ g(A) is s-wise t-intersecting]. (5) Then, if n>2k&t, the family g$(A) is a generating set for A. Proof. Since g(A) g$(A), we have A=U kn (g(A)) U kn (g$(A)). We have to show U kn (g$(A)) A. Let B # U kn (g$(A)). If A 1 , ..., A s&1 # A, there exist G 1 , ..., G s&1 # g(A), such that A 1 $G 1 , ..., A s&1 $G s&1 . Also, B$G, for G such that [G] _ g(A) is s-wise t-intersecting. So |A 1 & } } } & A s&1 & B| |G 1 & } } } & G s&1 & G| t. Therefore [B] _ A is in I s(n, k, t), and since A is maximal, we have that B # A. K In the next theorem, we show that the generating set g$(B) can be used to produce maximal families for larger n's. Theorem 2.8. Let n n 0 2k&t+1, and let B be a family in MI s(n 0 , k, t). Define A to be a family in I s(n, k, t) given by A=U kn (g$(B)). Then (1)
E(B)=E(A);
(2)
the family A is maximal in I s(n, k, t); and
(3)
g$(B)=g$(A).
Proof. Part (1). Given that B A, we see that E(B) E(A). So, we only need to show that E(A) E(B). Let e # E(A). Then, there exist A1 , ..., A s # A with e # A 1 & } } } & A s and |A 1 & } } } & A s | =t. By construction, Ai $G i for some G i # g$(B), for 1 i s. Using the fact that g$(B) is a generating set for B and Lemma 2.2, we get that |G 1 & } } } & G s | t. Therefore, we must have e # G1 & } } } & G s E(B). Part (2). Clearly A # I s(n, k, t). It is left to show that A is maximal. Let C # [1, n], with |C| =k and such that A _ [C] is s-wise t-intersecting. Considering that g$(B) is a generating set for A, we conclude that g$(B) _ [C] is a generating set for A _ [C]. Thus, by Lemma 2.2, g$(B) _ [C] is s-wise t-intersecting. Given that G # g$(B) G=E(B), we must have that g$(B) _ [C & E(B)] is s-wise t-intersecting. It is easy to prove that C & E(B) # g$(B). Indeed, let B to be any k-extension of C & E(B) contained in [1, n 0 ]. Thus, B _ [B] is s-wise t-intersecting, and since B is maximal, we have B # B. Considering that C & E(B) B & E(B) and that g$(B) _ [C & E(B)] is s-wise t-intersecting we conclude that C & E(B) # g$(B). Therefore, any k-extension of C & E(B) contained in [1, n] belongs to A, by definition; in particular, the set C # A.
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Part (3). First, we show that g$(B) g$(A). Let G # g$(B). Let A be any k-extension of G contained in [1, n]; by definition of A, it follows that A # A. Observing that G E(B)=E(A), we conclude that G A & E(A). Considering that [G] _ g(A) is a generating set for A, by Lemma 2.2, we must have [G] _ g(A) being s-wise t-intersecting. Thus, by the definition of g$(A), we have G # g$(A). Finally, we show that g$(A) g$(B). Let G$ # g$(A). Then, by definition, G$ A & E(A), for some A # A, and [G$] _ g(A) is s-wise t-intersecting. Considering that B A, we see that g(B) g(A). This implies that G$ B & g(B) is s-wise t-intersecting. Now, we just have to show that G$ B & g(B) for some B # B. Let B be any k-extension of A & E(A) contained in [1, n 0 ]. As G$ B, we know that B # A. This implies B is a k-extension of some G # g$(B), by construction of A. In addition, B [1, n 0 ], which implies B # B. We conclude the proof by observing that G$=G$ & E(B) B & E(B). K Now, we show that the generating set g$(A) for a family A is the same as a generating set g$(B) for some family B with smaller n, provided that n is large enough. Theorem 2.9. Let n0 max[2k&t+1, n(k, s, t)], and let A be a family in MI s(n, k, t), with n n 0 . Assume w.l.o.g. that E(A)=[1, |E(A)|]. Let B=U kn0 (g$(A)). Then (1)
E(B)=E(A);
(2)
the family B is maximal in I s(n 0 , k, t); and
(3)
g$(B)=g$(A).
Proof. Part (1). Since B A, it is clear that E(B) E(A). We have to show that E(A) E(B). Let e # E(A), then there exist sets A 1 , ..., A s # A with e # A 1 & } } } & A s and |A 1 & } } } & A s | =t. Let us call I=A 1 & } } } & A s . Consider G i =A i & E(A), for 1 i s. Then G 1 , ..., G s # g(A), and I G 1 & } } } & G s . Thus |G i | |I | =t. Moreover, G 1 , ..., G s # g$(A), so U kn0 ([G 1 , ..., G s ]) B. We choose the following k-extensions of G 1 , ..., G s contained in [1, n 0 ]: v B 1 =G 1 _ L 1 , for some L 1 # [1, n 0 ] "(G 1 _ (G 2 & } } } & G s )); v B i =G i _ L i , for some L i [1, n 0 ] "(G i _ B 1 ), i=2, ..., s. There exists always such L 1 , since |[1, n 0 ] "(G 1 _ (G 2 & } } } & G s ))| n 0 & (|G 1 | + |G 2 & } } } & G s | & |I | ) 2k&t+1&(|G 1 | +k&t) k& |G 1 | . For 2 i s, there exists always such L i , since |[1, n 0 ] "(G i _ B 1 )| =n 0 & (|G i | + |B 1 | & |G i & B 1 | ) 2k&t+1& |G i | &k+t k& |G i | .
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Therefore B 1 , ..., B s # B and B 1 & } } } & B s =(G 1 _ L 1 ) & (G 2 _ L 2 ) & } } } & (G s _ L s ) =(G 1 _ L 1 ) & G 2 & } } } & G s
(by the choice of L 2 , ..., L s )
=(G 1 & G 2 & } } } G s ) _ (L 1 & G 2 & } } } & G s ) =I. So e # B 1 & } } } & B s and |B 1 & } } } & B s | = |I| =t. Therefore e is essential for B. Part (2). One can easily check from B=U kn0 (g$(A)) and G # g$(A) G [1, |E(A)| ] [1, n 0 ], that B # I s(n 0 , k, t). We only have to prove that B is maximal. Let C # [1, n 0 ] such that B$=B _ [C] is s-wise t-intersecting. Then, g = g$(A) _ [C] is a generating set of B$. By Lemma 2.2, the set g is s-wise t-intersecting, which implies that for any G 1 , ..., G s&1 # g$(A), we have |G 1 & } } } & G s&1 & C | t. So, A _ [C] is s-wise t-intersecting, and since A is maximal we must have C # A. Therefore, C & E(A) # g$(A), which implies C # B. Part (3). Let us first prove that g$(A) g$(B). Let G # g$(A). Then, G A & E(A), for some A # A, and g(A) _ [G ] is s-wise t-intersecting. Since B A and E(B)=E(A), we have that g(B) g(A). Thus, g(B) _ [G ] is s-wise t-intersecting. In addition, B=G _ L # B, for any L [1, n 0 ] "G. Then, B & E(B)=(G _ L) & E(A)=G _ (L & E(A)). So, G B & E(B) for some B # B, and g(B) _ [G ] is s-wise t-intersecting. Therefore, G # g$(B). We must show that g$(B) g$(A). Let G # g$(B). Thus, G B & E(B), for some B # B, and g(B) _ [G ] is s-wise t-intersecting. Given that B A and E(B)=E(A), we conclude that G B & E(B)=B & E(A), with B # A. Let G 1 , ..., G s&1 # g(A). Considering that g(A) g$(A) g$(B), we have that G 1 , ..., G s&1 # g$(B). Using the fact that g$(B) is a generating set of B and n 0 >2k&t, by Lemma 2.2, we conclude that g$(B) is s-wise t-intersecting, and in particular that |G & G 1 & } } } & G s&1 | t. Therefore, g(A) _ [G ] is s-wise t-intersecting, which implies G # g$(A). K The following corollary implies that the solution of Problem 1 for k, s, t and n n 0 (k, s, t) can be obtained from the solution of Problem 1 for k, s, t and n=n 0 (k, s, t). Corollary 2.10. Let G(n)=[g$(A) : A # MI s(n, k, t)] and n 0 (k, s, t) =max[n(k, s, t), 2k&t+1]. If n 1 , n 2 n 0 (k, s, t), then G(n 1 ) is isomorphismequivalent to G(n 2 ). Proof. Assume w.l.o.g. that n 2 n 1 . Theorem 2.9 gives that for every family A # MI s(n 2 , k, t), there exist a family B # MI s(n 1 , k, t) such that
MAXIMAL INTERSECTING SET FAMILIES
63
g$(B)t g$(A). Theorem 2.8 gives that for every family B # MI s(n 1 , k, t), there exist a family A # MI s(n 2 , k, t) such that g$(A)= g$(B). K Remark 2.11. For A # MI s(n, k, t), the generating set g$(A) is not necessarily minimal in the sense of set inclusion. For representing A, we may also use the generating set g"(A)=[G # g$(A) : for any G 1 # g$(A), G 1 G implies G 1 =G]. It is easy to check that g"(A) is a generating set of A, and that g"(A) satisfies the same properties as g$(A), given in Theorem 2.9 and Corollary 2.10. Example 2.12.
Let A # MI 2(8, 4, 1) as given in Example 2.6, with
g(A)=[[1, 2, 3, 4], [1, 5, 6, 7], [3, 5, 6, 7], [4, 5, 6, 7]] _ U 37 ([[2, 5], [2, 6], [2, 7]]) _ U 47 ([[2, 5], [2, 6], [2, 7]]). Then, g$(A)= g(A) _ [[2, 5], [2, 6], [2, 7]]], and g"(A)=[[1, 2, 3, 4], [1, 5, 6, 7], [3, 5, 6, 7], [4, 5, 6, 7], [2, 5], [2, 6], [2, 7]].
3. A CONSTRUCTION OF FAMILIES IN MI s(n+1, k+1, t+1) FROM FAMILIES IN MI s((n, k, t) In this section, we show how a family in MI s(n, k, t) can be used to construct a family in MI s(n+1, k+1, t+1), for any s, n, k, and t. We also show that if the construction is applied to two non-isomorphic families, then the resulting families are also non-isomorphic (Theorem 3.5). In addition, for n>2k&t, this construction increases the number of essential numbers by exactly one (Proposition 3.8). The idea behind the construction is as follows. Let A # MI s(n, k, t). The family A(n+1) obtained by adding the element n+1 to every set in A is obviously a family in I s(n+1, k+1, t+1), but not necessarily maximal. The construction adds appropriate sets to A(n+1) in order to obtain a family B that is maximal. In fact, we show that B is the only maximal family in MI s(n+1, k+1, t+1) that contains A(n+1) .
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Construction 3.1. Let A be a family in MI s(n, k, t). Let A(n+1) = [A _ [n+1] : A # A]. For any A # A, denote E(A)=[(C 1 , ..., C s&1 ) # A : |C 1 & } } } & C s&1 & A| =t], and L(A)=( (C1 , ..., Cs&1 ) # E(A) (C 1 & } } } & C s&1 )) "A. Let, for any A # A, A*(A)=
{
[A _ [l ] : l # L(A)] [A _ [i ] : i # [1, n] "A]
if E(A){<, if E(A)=<,
and A*= A # A A*(A). Define B to be the family A(n+1) _ A*. Theorem 3.2. Let A be a family in MI s(n, k, t). Then the family B=A(n+1) _ A* given by Construction 3.1 is a family in MI s(n+1, k+1, t+1). Moreover, family B is the only family in MI s(n+1, k+1, t+1) that contains the subfamily A(n+1) . Proof. First, we will prove thnat B # I s(n+1, k+1, t+1). It is clear that B is (k+1)-uniform, so it remains to prove that B is s-wise (t+1)intersecting. Let B 1 , ..., B s # B, we need to show that |B 1 & } } } & B s | t+1. We split the analysis in two cases. (1)
If B 1 , ..., B s # A(n+1) , then it is trivial that |B 1 & } } } & B s | t+1.
(2) If for some i, 1 i s, B i # A*, we assume w.l.o.g. that B 1 # A*. We know that there exist A 1 , ..., A s # A such that B i $A i , for 1 i s. If |A 1 & } } } & A s | t+1, then we are done. This include the cases when E(A 1 )=<, for in such case |A 1 & C 1 & } } } & C s&1 | t+1, for any C 1 , ..., C s&1 # A. When |A 1 & } } } & A s | =t, we have E(A 1 ){<, and by the construction of A*(A 1 ), we must have B 1 =A 1 _ [l 1 ], with l 1 # L(A 1 ). But, by the construction of L(A 1 ), we must have l 1 # (A 2 & } } } & A s ) "A 1 then |B 1 & } } } & B s | |A 1 & } } } & A s | +1=t+1. We must show that B is a maximal family in I(n+1, k+1, t+1), and B is the only maximal family in I(n+1, k+1, t+1) that contains the subfamily A(n+1) . Let S [1, n+1] be an arbitrary (k+1)-set that is s-wise (t+1)-intersecting with any s&1 sets B 1 , ..., B s&1 # A(n+1) . We prove both of the above statements by showing that S # B. If n+1 # S, then S "[n+1] must be s-wise t-intersecting with any s&1 sets A 1 , ,..., A s&1 # A. Since A is maximal, S "[n+1] # A, and so S # B. If n+1 Â S, then let a # S and call S a =S"[a]. Thus [S] _ A is s-wise (t+1)-intersecting, which implies that [S a ] _ A is s-wise t-intersecting, and then S a # A. Now, if E(S a )=<, by construction of A*(S a ), we have S=S a _ [a] # B. Otherwise, if E(S a ){<, for any (C 1 , ..., C s&1 ) # E(S a ), we have |C 1 & } } } & C s&1 & S a | =t, but |C 1 & } } } & C s&1 & S| t+1, and so we must have a # C 1 & } } } & C s&1 . Therefore a # L(S a ), and then S= Sa _ [a] # B. K
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Example 3.3. Let A # MI 2(6, 3, 1) given by the first column of the table. The set B=A(7) _ A* is in MI 2(7, 4, 2). A
E(A)
[1, 2, 3]
<
[1, 2, 4] [1, 2, 5] [1, 2, 6] [1, 3, 4] [1, 3, 5] [1, 3, 6] [2, 3, 4] [2, 3, 5] [2, 3, 6]
[1, 3, 5] [1, 3, 4] [1, 3, 4] [1, 2, 5] [1, 2, 4] [1, 2, 4] [1, 2, 5] [1, 2, 4] [1, 2, 4]
[1, 3, 6] [1, 3, 6] [1, 3, 5] [1, 2, 6] [1, 2, 6] [1, 2, 5] [1, 2, 6] [1, 2, 6] [1, 2, 5]
[2, 3, 5] [2, 3, 4] [2, 3, 4] [2, 3, 5] [2, 3, 4] [2, 3, 4] [1, 3, 5] [1, 3, 4] [1, 3, 4]
[2, 3, 6] [2, 3, 6] [2, 3, 5] [2, 3, 6] [2, 3, 6] [2, 3, 5] [1, 3, 6] [1, 3, 6] [1, 3, 5]
L(A)
A*(A)
[3] [3] [3] [2] [2] [2] [1] [1] [1]
[1, 2, 3, 4] [1, 2, 3, 6] [1, 2, 3, 4] [1, 2, 3, 5] [1, 2, 3, 6] [1, 2, 3, 4] [1, 2, 3, 5] [1, 2, 3, 6] [1, 2, 3, 4] [1, 2, 3, 5] [1, 2, 3, 6]
A(7) [1, 2, 3, 5]
[1, 2, 3, 7] [1, 2, 4, 7] [1, 2, 5, 7] [1, 2, 6, 7] [1, 3, 4, 7] [1, 3, 5, 7] [1, 3, 6, 7] [2, 3, 4, 7] [2, 3, 5, 7] [2, 3, 6, 7]
Now we introduce some notation that will be used in the next theorem, as well as in the sections to follow. Notation 3.4. For any family B of subsets of [1, n], and for any x, y, z # [1, n], we denote Bx =[B # B : x # B], Ba, b =Ba & Bb ,
for
a, b # [x, x, y, y ],
Bx =B "Bx , Ba, b, c =Ba & Bb & Bc ,
for
a, b, c # [x, x, y, y, z, z ].
The following theorem shows that if Construction 3.1 is applied to nonisomorphic families, then the resulting families are also non-isomorphic. Theorem 3.5. Let A 1, A 2 # MI s(n, k, t), and let B 1 and B 2 be the families in MI s(n+1, k+1, t+1) given by the application of Construction 3.1 to the families A 1 and A 2, respectively. Then B 1 tB 2 implies A 1 tA 2. Proof. Let ? be the permutation on [1, n+1] such that B 2 =?(B 1 ). If ?(n+1)=n+1, then trivially A 2 =?(A 1 ). Otherwise, we can assume w.l.o.g. that ?(n+1)=n. Let us call B=B 2 =?(B 1 ). For a family C we denote C "[x]=[C "[x] : C # C]; we can then write ?(A 1 )=Bn, n+1 "[n] _ Bn, n+1 "[n] A 2 =Bn, n+1 "[n+1] _ Bn, n+1 "[n+1]. We want to show ?(A 1 )tA 2. It is clear that Bn, n+1 "[n]t Bn, n+1 "[n+1). It is enough to prove that Bn, n+1 "[n]=Bn, n+1 "[n+1].
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We claim that for any A # Bn, n+1 "[n], the set [A ] _ A 2 is s-wise t-intersecting. By definition, A 2 is s-wise t-intersecting. Now, let A 1 , ..., A s&1 # A 2 and let B i =A i _ [n+1], for 1 i s&1, B =A _ [n]. Observe that B 1 , ..., B s&1 , B # B. Let us call I A =A 1 & } } } & A s&1 & A and I B =B 1 & } } } & B s&1 & B. First, we observe that I A =I B"[n, n+1], and since B # Bn, n+1 , we known that n+1 Â I B . So |I A | = |I B"[n]| (t+1)&1=t. So, [A ] _ A 2 is s-wise t-intersecting. Then, since A 2 is maximal it follows that A # A 2. Therefore, Bn, n+1"[n] A 2, which means Bn, n+1"[n] Bn, n+1"[n+1]. By an analogous argument, we show that Bn, n+1"[n+1] Bn, n+1"[n]. Therefore, Bn, n+1"[n+1]=Bn, n+1"[n], which completes the proof. K An immediate consequence of the previous theorem is that, for any s 2, the number of non-isomorphic families is non-decreasing, when parameters n, k and t are increased by one. Recall the meaning of D(}) from Notation 1.2. Corollary 3.6. For any s, n, k and t 1, we have D(MI s(n+1, k+1, t+1)) D(MI s(n, k, t)). The following lemma is a well-known result for s=2 (see, e.g., [6]), saying basically that, for n 2k&t, if A # MI s(n, k, t), then A Â MI s(n, k, t+1). Lemma 3.7.
Let n 2k&t and A # MI s(n, k, t). Then E(A){<.
Now we show that when we apply the previous construction for n 2k&t, then the number of essential numbers increases by exactly one. Proposition 3.8. Let n 2k&t. Let A be a family in MI s(n, k, t), and let the family B # MI s(n+1, k+1, t+1) be the family given by Construction 3.1. Then E(B)=E(A) _ [n+1]. Sketch of the Proof. Lemma 3.7 is used to prove the inclusion E(A) _ [n+1] E(B). The inclusion E(B) E(A) _ [n+1] is proven by showing that for any essential s-tuple for e in B, e{n+1, one can carefully remove one element from each set of the s-tuple producing an essential s-tuple for e in A. K 4. ALL MAXIMAL k-UNIFORM PAIRWISE t-INTERSECTING FAMILIES FOR k t+2 In this section, we solve Problem 1 for the case of s=2 and k=t+1, t+2 (see Fig. 1). The case of k=t+1 is simple, but we include it here for the sake of completeness.
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67
FIG. 1. Determination of all maximal families for s=2, k=t+2, for all t 1 (see Theorem 4.8).
4.1. Determination of All Families in MI 2(n, t+1, t) In this subsection, we determine all families A # MI 2(n, t+1, t) for arbitrary n t+2, and show that n(t+1, 2, t)=t+2, for all t 1. We will use the following notation, for any k and n: ( [n] k )= [B [1, n] : |B| =k]. Proposition 4.1. following hold: (1)
Let t 1 and n t+2, and A # MI 2(n, t+1, t). The
if n=t+2, then A=( [t+2] t+1 );
[n] ) : B$[1, t]] or (2) if n t+3, then either AtB$=[B ( t+1 ). At( [t+2] t+1
File: 582A 294616 . By:XX . Date:09:06:99 . Time:15:42 LOP8M. V8.B. Page 01:01 Codes: 1177 Signs: 550 . Length: 45 pic 0 pts, 190 mm
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Sketch of the Proof. Part (1). The set ( [t+2] t+1 ) containing every possible subset is t-intersecting. Part (2). The proof consists of a simple case analysis. In the case that there exist distinct sets A 1 , A 2 , A 3 # A with |A 1 & A 2 & A 3 | =t it is easy to show that AtB$. In the complementary case, we conclude that sets in A must be contained in a fixed (t+2)-set. K Thus, by Part (1), we have At( [t+2] t+1 ). Corollary 4.2. For t 1, we have n(t+1, 2, t)=t+2. Proof. From Corollary 2.5, we have n(t+1, 2, t)=E max (t+1, 2, t), but by Proposition 4.1 we have E max (t+1, 2, 5)=t+2. K 4.2. Determination of All Families in MI 2(n, t+2, t) In this subsection, we show that Construction 3.1 is enough to generate all the sets in MI 2(n, t+3, t+1) when applied to all the sets in MI 2(n, t+2, t), for any n and t 2 (Theorem 4.5). We also show that n(t+2, 2, t)=t+6 (Theorem 4.7). Using these two results combined with results from previous sections, we manage to reduce Problem 1 (for any s=2, n, t and k=t+2) to the solution of Problem 1 for (s, n, k, t) # [(2, 7, 3, 1), (2, 8, 4, 2)] (Theorem 4.8). This solution is obtained by computer and shown in Table I. Throughout this section, we will largely use Notation 3.4. In addition, for any A # I s(n, k, t), and a # [1, n], we denote by n a the number of sets in A containing a, that is n a = |Aa | . Lemma 4.3. Let A be a family in MI 2(n, t+2, t), for t 3. Let x # [1, n] such that n x =max i # [1, n] n i . Then the family Ax is pairwise (t+1)-intersecting. Proof. It is clear that Ax is pairwise t-intersecting, since Ax A. In order to prove that Ax is, in addition, pairwise (t+1)-intersecting, we assume by contradiction that there exist A 1 , A 2 # Ax , such that |A 1 & A 2 | =t. Then A 1 and A 2 must be of the form A 1 =[ y 1 , y 2 , ..., y t , : 1 , ; 1 ],
and
A 2 =[ y 1 , y 2 , ..., y t , : 2 , ; 2 ],
for distinct y 1 , y 2 , ..., y t , : 1 , ; 1 , : 2 , ; 2 , different from x, For any i, since n yi n x we must have |Ayi , x | |Ayi , x | 2.
(6)
We claim that Ayi , x cannot be pairwise (t+1)-intersecting. Indeed, if it was so, it would imply that the family obtained by replacing x by y i in Ayi , x , denoted by Ayi , x &[x]+[ y i ], would t-intersect all other sets in A.
MAXIMAL INTERSECTING SET FAMILIES
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Then, since A is maximal, we would have Ayi , x &[x]+[y i ] A. Consequently, we would get |Ayi , x | |(Ayi , x &[x]+[ y i ]) _ [A 1 ] _ [A 2 ]| = |Ayi , x | +2> |Ayi , x | , contradicting (6). Therefore, for i=1, ..., t, there exist B i1 , B i2 # Ayi , x , with |B i1 & B i2 | =t.
(7)
Since B i1 , B i2 # Ayi , x and must also t-intersect both A 1 and A 2 , they must be of the form B i1 =[x, y 1 , y 2 , ..., y t ]"[ y i ] _ Z i1 ,
B i2 =[x, y 1 , y 2 , ..., y t ]"[ y i ] _ Z i2 ,
with Z i1 , Z i2 # [[: 1 , : 2 ], [: 1 , ; 2 ], [; 1 , : 2 ], [; 1 , ; 2 ]]. Now, condition (7) translates into Z i1 & Z i2 =<, for all i=1, ..., t, which implies [Z i1 , Z i2 ] equals either [[: 1 , : 2 ], [; 1 , ; 2 ]] or [[: 1 , ; 2 ], [; 1 , : 2 ]]. Since t 3, by i i i i the pigeon-hole principle for some i 1 {i 2 , we have [Z 11, Z 21 ]=[Z 12, Z 22 ]. i1 i2 i1 i2 So, w.l.o.g., Z 1 =Z 1 , which implies |B 1 & B 2 | =t&1
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B & L${<. Indeed, let A$ # A$. Then |A$| =t+1, |A$ & B| t 3 and |A$ & L$| t&1. Then, |B & L$| |A$ & B & L$| |A$ & B| + |A$ & L$| & |A$| t+(t&1)&(t+1)=t&2 1. Let y # B & L$. By Lemma 4.4, (B"[ y] _ [x]) # A and so (B"[ y]) # A$. By definition of L$, we have |(B "[ y]) & L$| t&1. Since y # B & L$, we get |B & L$| t, and finally |B & L| t. K We examine the consequences of the previous theorem on the size of kernels for s=2 and t=k&2 2. Corollary 4.6. Let t=k&2 2. Then n(t+2, 2, t)=n(4, 2, 2)+t&2. Proof. Let n t+5 and let B # MI 2(n, t+2, t). By Theorem 4.5, there exists A # MI 2(n&(t&2), 4, 2) such that each B is obtained by t&2 applications of Construction 3.1 to A. Thus, by Proposition 3.8, we have |E(B)| = |E(A)| +t&2. Finally, using Corollary 2.5, we get n(t+2, 2, t)= n(4, 2, 2)+t&2. K Finally, we just need to determine n(3, 2, 1) and n(4, 2, 2) in order to obtain n(t+2, 2, t) for any t 1. Theorem 4.7.
For any t 1, we have n(t+2, 2, t)=t+6.
Proof. First, we observe that n(3, 2, 1) 7 and n(4, 2, 2) 8, since families G 8 # MI 2(7, 3, 1) and H 8 # MI 2(8, 4, 2) (see Table I) are such that E(G 8 )=7 and E(H 8 )=8. In addition, if n(4, 2, 2) 8, using that n(k&1, s, t&1) n(k, s, t)&1, we get that n(3, 2, 1) 7. This implies n(3, 2, 1)=7 and n(4, 2, 2) = 8, which combined with Corollary 4.6, leads to n(t+2, 2, t)=t+6, for t 1. Thus, it is enough to show that n(4, 2, 2) 8. Let A # MI 2(n, 4, 2) for arbitrary n 8, and let A, B # A be an essential pair for A (there exists such a pair by Lemma 3.7). We can assume w.l.o.g. that A=[1, 2, 3, 4] and B=[1, 2, 5, 6]. We can also assume that A has no set of the form [1, 2, a, b] with a, b # [1, n] " [1, 6], for this would imply that [1, 2]. We can also assume that A has no set of the form [1, 2, a, b] with a, b # [1, n] " [1, 6], for this would imply that [1, 2] must be contained in every other set in A and then |E(A)| =2. Therefore, the remaining sets that are possibly contained in A and that include an element x  [1, 6] are of the form [1, 2, a, x],
for
a # [3, 4, 5, 6]
[b, c, e, x],
for
b # [1, 2], [c, e] [[3, 5], [3, 6], [4, 5], [4, 6]], and x # [7, n].
and
x # [7, n];
(8)
(9)
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Therefore, any essential element x # [7, n] must have an essential pair chosen from sets in (8) and (9). We will show there are at most 2 such essential elements, which implies |E(A)| 8. Define a graph G which vertices corresponding to the possible triples accompanying an essential element x # [7, n], given by (8) and (9), and edges placed whenever the triples corresponding to its endings intersect. Colour the edges whose triples corresponding to its endings intersect in exactly 1 element. Now, for each essential element x # [7, n] for A, let e(x) be an edge of G connecting some triples T1 and T2 such that T1 _ [x] and T2 _ [x] is an essential pair for A. Let E$=[e(x) : x # E(A) " [1, 6]]. Let G$ be the subgraph of G induced by the set of all vertices that are endings for some edge in E$. Since A is 2-intersecting, G$ is a complete graph whose coloured edges form a matching. We will show that such E$ must have |E$| 2, which implies |E(A)| 8. We reduce the size of the subgraphs we have to look at, by observing that another essential pair for x # [7, n], must be equivalent to one of the following cases: C 1 =[1, 3, 5, x ], D 1 =[1, 2, 4, x ]; C 2 =[1, 3, 5, x ], D 2 =[1, 4, 6, x ]; or C 3 =[1, 3, 5, x ], D 3 =[2, 3, 6, x ]. For i=1, 2, 3, we analyse the case of e(x ) joining v Ci and v Di , the vertices corresponding to C i and D i , respectively. Let Gi be the subgraph of G that is induced by v Ci , v Di and every vertex v that have non-coloured edges incident to both v Ci and v Di . The edges in E$ are coloured and are contained in Gi for some 1 i 3. In Fig. 2, we show graphs G1 , G2 and G3 with thick edges representing the coloured edges. By inspection, we conclude that |E$ | 2. K The following theorem is a combination of many of the previous results. Theorem 4.8. The non-isomorphic families in MI 2(8, 4, 2) and in MI 2 (7, 3, 1) completely determine all non-isomorphic families in MI 2(n, t+2, t), for all n t+6 and t 1.
FIG. 2.
Coloured edges of graphs G1 , G2 , and G3 .
File: 582A 294620 . By:XX . Date:09:06:99 . Time:15:42 LOP8M. V8.B. Page 01:01 Codes: 2805 Signs: 1961 . Length: 45 pic 0 pts, 190 mm
72
LUCIA MOURA
Proof. Suppose we have listed all non-isomorphic families in MI 2(8, 4, 2) and MI 2(7, 3, 1). We outline how to obtain all nonisomorphic sets in MI 2(n, t+2, t) for arbitrary n t+6, t 1. If t=1, by our assumptions, we have listed all non-isomorphic families in MI 2(t+6, t+2, t)=MI 2(7, 3, 1). If t 2, perform t&2 applications of Construction 3.1 to each of the non-isomorphic sets in MI 2(8, 4, 2). By Theorem 3.5 and Theorem 4.5, this gives all non-isomorphic sets in MI 2(8+(t&2), 4+(t&2), 2+(t&2))=MI 2(t+6, t+2, t). Now, by Theorem 4.7, we have that n 0 (t+2, t)=t+6. Corollary 2.10 implies that computing generating sets g$(A), as defined in (5), for all non-isomorphic sets in MI 2(t+6, t+2, t) gives generating sets for all non-isomorphic sets in MI 2(n, t+2, t), for n t+6. K Corollary 4.9. Let t 1, and n t+6. Then, D(MI 2(n, t+2, t))=
{
D(MI 2(7, 3, 1))=15, D(MI 2(8, 4, 2))=17,
if t=1, if t 2.
A complete list of non-isomorphic sets in MI 2(8, 4, 2) and in MI 2(7, 3, 1) is given in Table I.
ACKNOWLEDGMENTS This article is part of my Ph.D. research at University of Toronto under the supervision of Professor Rudi Mathon. I thank my supervisor for the helpful discussions during the preparation of this manuscript. I also thank an anonymous referee for valuable suggestions.
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