Intersecting k -uniform families containing a given family

Discrete Mathematics 340 (2017) 140–144

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Intersecting k-uniform families containing a given family Jun Wang a , Huajun Zhang b, * a b

Department of Mathematics, Shanghai Normal University, Shanghai 200234, PR China Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR China

article

a b s t r a c t

info

A family A of sets is said to be intersecting if A ∩ B ̸ = ∅ for any A, B ∈ A. Let m, n, k and r be positive integers with m ≥ 2k ≥ 2r > n ≥ k. A family F of sets is called

Article history: Received 19 September 2014 Received in revised form 29 July 2016 Accepted 31 July 2016

([m]) (m, n, k, r)-intersecting family if F is an intersecting subfamily of k containing {A ∈ (an [m]) : |A ∩ [n]| ≥ r }. Maximum (m, k, k, k)- and (m, k + 1, k, k)-intersecting families k

were determined by the well-known Erdős–Ko–Rado theorem and Hilton–Milner theorem, respectively. Recently, Li et al. determined maximum (m, n, k, k)-intersecting family when n = 2k − 1, 2k − 2, 2k − 3 or m is sufficiently large. In this paper, we determine all the maximum (m, n, k, r)-intersecting families. © 2016 Elsevier B.V. All rights reserved.

Keywords: Family of sets Intersecting family Erdős–Ko–Rado theorem Hilton–Milner theorem

1. Introduction A finite set X is called an n-set if its cardinality |X | equals n. In particular, let [n] denote the n-set {1, . . . , n} and for a positive integer k ≤ n, define [k, n] = {k, k + 1, . . . , n}. A(family of subsets of X is said to be intersecting if no two members ) (|X |) X are disjoint. The family of all k-subsets of X is denoted by k , whose cardinality is the well-known binomial coefficient k . ([n]) We are interested in the intersecting subfamilies of k . Here, a maximum intersecting family is an intersecting family of ([n]) ([n]) the largest size. Note that k is intersecting if n < 2k. While for n = 2k, an intersecting subfamily F of k contains at most one of A ∈

([n]) k

and its complement A = [n] \ A. So a maximum intersecting subfamily of

([n])

([2k]) k

has size

1 2k 2 k

( )

=

(2k−1) k−1

.

If all members of a family F ⊂ k contain a fixed element, say t ∈ [n], then F is obviously an intersecting family and is said to be a star, or a t-star ([n]if) the element t needs (n−1to ) be indicated. A star is said to be a trivial intersecting family. A trivial intersecting subfamily of k can have at most k−1 members. The general case is solved by Erdős, Ko, and Rado [3], whose

([n])

(n−1)

pioneering result states that if F is an intersecting subfamily of k , then |F | ≤ k−1 subject to n ≥ 2k, and equality holds if and only if F is a star, except for n = 2k. This theorem was reproved many times, and initiated a new field of combinatorial set theory. The reader is referred to Deza and Frankl [2], Frankl [4], Borg [1] for( surveys on relevant results. Hilton and ([nand ) (n− ) ]) n−1 k−1 Milner [6] proved that if F is a nontrivial intersecting subfamily of k , then |F | ≤ k−1 − k−1 + 1, and equality implies

([n])

that F consists of a fixed A ∈ k together with a t-star S such that t ̸ ∈ A and every member of S intersects A, except for ([k+1]) k = 3. By symmetry we may assume that t = 1 and A = [2, k + 1]. Then F contains . k In [7], Li et al. introduced a generalization of the above structure, which we state in a more general form as follows. Let m, n, k and r be positive integers with m ≥ 2k > n ≥ k ≥ r > 0, and set A(m, n, k, r) =

*

{ A∈

( ) [m ] k

} : |A ∩ [n]| ≥ r .

Corresponding author. E-mail addresses: [email protected] (J. Wang), [email protected] (H. Zhang).

http://dx.doi.org/10.1016/j.disc.2016.07.027 0012-365X/© 2016 Elsevier B.V. All rights reserved.

J. Wang, H. Zhang / Discrete Mathematics 340 (2017) 140–144

Clearly, A(m, n, k, k) =

, A(m, n, k, r) =

([n])

([m])

k

k

∑k (n)(m−n)

141

if m = n + k − r, and the size of A(m, n, k, r), written as f (m, n, k, r),

equals i=r i k−i . 2r > n, A(m, n, k, r) is an intersecting family. We call a family F (m, n, k, r)-intersecting if A(m, n, k, r) ⊆ F ⊆ ([m]When ) . Let α (m, n, k, r) denote the largest size of (m, n, k, r)-intersecting families. k Let F be an (m, n, k, r)-intersecting family. For every A ∈ F \ A(m, n, k, r), if |A ∩ [n]| ≤ n − r, then A would not intersect some B ∈ A(m, n, k, r). Therefore, n − r + 1 ≤ |A ∩[n]| ≤ r − 1. From this it is seen that if n = 2r − 1, then F = A(m, n, k, r), which is a trivial case. Suppose n ≤ 2r − 2 and set B(m, n, k, r) =

{ A∈

( ) [m ]

}

: n − r + 1 ≤ |A ∩ [n]| ≤ r − 1 k = A(m, n, k, n − r + 1) \ A(m, n, k, r).

(1)

Then F ⊆ A(m, n, k, r) ∪ B(m, n, k, r). Let β (m, n, k, r) denote the size of maximum intersecting subfamilies of B(m, n, k, r). Then it is easily seen that

α (m, n, k, r) = f (m, n, k, r) + β (m, n, k, r). Our tasks are indeed to determine β (m, n, k, r) and the intersecting families of size β (m, n, k, r). For any t ∈ [n], set Bt (m, n, k, r) = {A ∈ B(m, n, k, r) : t ∈ A}, and define Ht (m, n, k, r) = A(m, n, k, r) ∪ Bt (m, n, k, r).

Then Ht (m, n, k, r) is an (m, n, k, r)-intersecting family of size h(m, n, k, r) =

) k ( )( ∑ n m−n k−i

i

i=r

+

( )( ) r −1 ∑ n−1 m−n . i−1 k−i i=n−r +1

Consider the special case when n = 2r − 2:

) ( )} [2r − 2] [2r − 1, m] ,B ∈ . r −1 k−r +1 { ([2r −2]) ([2r −1,m])} For any maximum intersecting subfamily T of r −1 , set BT (m, 2r − 2, k, r) = A ∪ B : A ∈ T , B ∈ k−r +1 , and define B(m, 2r − 2, k, r) =

{

(

A∪B:A∈

HT (m, 2r − 2, k, r) = A(m, 2r − 2, k, r) ∪ BT (m, 2r − 2, k, r).

Then HT (m, 2r − 2, k, r) is an (m, 2r − 2, k, r)-intersecting family of size h(m, 2r − 2, k, r). Theorem 1.1. Let m, n, k and r be positive integers with m ≥ 2k ≥ 2r ≥ n + 2 ≥ k + 2. If F is an (m, n, k, r)-intersecting family, then |F | ≤ h(m, n, k, r), and equality holds ([2r −if2]and ) only if F = Ht (m, n, k, r) for some t ∈ [n], or F = HT (m, 2r − 2, k, r) for some maximum intersecting family T in r −1 , except for m = 2k. Note that, in [7], Li et al. dealt with the case r = k. They proved that the result of Theorem 1.1 holds for n = 2k − 1, 2k − 2, 2k − 3 or m sufficiently large, and asked whether or not it is always the case. Theorem 1.1 has answered their question in a more general setting. Our tool is the shift operation, which was introduced by Erdős, Ko and Rado in the original paper [3] (see also [4]). We shall review it in the next section, and present some preliminary results in Section 3. The proof of Theorem 1.1 is given in Section 4. 2. Shift operation For a family F ⊆

{ sij (F ) =

([m]) k

and i, j ∈ [m] with i < j, define

(F \ {j}) ∪ {i}, F,

if j ∈ F , i ̸ ∈ F , (F \ {j}) ∪ {i} ̸ ∈ F ; otherwise

for each F ∈ F , and sij (F ) = {sij (F ) : F ∈ F }. Clearly, |sij (F )| = |F |, and it was proved in [3] that sij (F ) is an intersecting family if F is so. Apply this procedure to F until we obtain a family H such that sij (H) = H for ([many ) 1 ≤ i < j ≤ m. Such an H is called compressed. ] Now, let F be an intersecting subfamily of k satisfying sim (F ) = F for each i < m. Then F has a canonical decomposition F0 ∪ F1 , where F0 = {F ∈ F : m ̸ ∈ F } and F1 = {F ∈ F : m ∈ F }. Then F0 is an intersecting ([mF−1= ]) subfamily of . A key point that was proved in [3] is that the family F1′ = {F \ {m} : F ∈ F1 } is an intersecting k subfamily of

([m−1]) k−1

if m > 2k.

142

J. Wang, H. Zhang / Discrete Mathematics 340 (2017) 140–144

3. Preliminary results Recall that

{

A(m, n, k, r) =

( ) [m ]

A∈

k

} : |A ∩ [n]| ≥ r ,

which is of size f (m, n, k, r) =

) k ( )( ∑ n m−n k−i

i

i =r

.

For any t ∈ [n], define At (m, n, k, r) = {A ∈ A(m, n, k, r) : t ∈ A}. Then At (m, n, k, r) is an intersecting subfamily of A(m, n, k, r), which is of size f (m, n, k, r) = ∗

)( ) k ( ∑ n−1 m−n i−1

i =r

k−i

.

Lemma 3.1. Let m, n, k (nand ) r be positive integers with m (m≥ ) n ≥ k ≥ r > 0, n ≥ k + r and m − n ≥ k − r. Then (1) f (m, n, k, k) = k and f (m, m − k + r , k, r) = k ; and for m > n, (2) f ∗ (m, n, k, r) = f ∗ (m − 1, n, k, r) + f ∗ (m − 1, n, k − 1, r); (3) h(m, n, k, k) = h(m − 1, n, k, k) + f ∗ (m − 1, n, k − 1, n − k + 1); (4) h(m, n, k, r) = h(m − 1, n, k, r) + h(m − 1, n, k − 1, r) where k > r. Proof. (1) is obtained by the following simple observation: A(m, n, k, r) = k if r = k and A(m, n, k, r) = if k m = n + k − r. (2)–(4) can be directly verified in a similar way. Here we show only (3) and leave the others to the reader.

([n])

([m])

( )( ) k−1 ∑ n−1 m−n k i−1 k−i i=n−k+1 ( ) ( ) [( ) ( )] k−1 ∑ n n−1 m−1−n m−1−n = + + k i−1 k−i k−i−1 i=n−k+1 ( ) ( )( ) ( )( ) k−1 k−1 ∑ ∑ n n−1 m−1−n n−1 m−1−n = + + k i−1 k−i i−1 k−i−1 i=n−k+1 i=n−k+1

h(m, n, k, k) =

( ) n

+

= h(m − 1, n, k, k) + f ∗ (m − 1, n, k − 1, n − k + 1).

□

Lemma 3.2. Let m, n, k and r be positive integers with m ≥ n ≥ k ≥ r > 0, n ≥ k + r and m > n + k − r. If F is an intersecting subfamily of A(m, n, k, r) with sim (F ) = At (m; n; k; r) for some t ∈ [n] and i < m, then F = At (m; n; k; r). Proof. Note that m > n ≥ t. Suppose i ̸ = t. By definition, for each A ∈ A(m; n; k; r), A ∈ At (m; n; k; r) if and only if t ∈ A, or equivalently, t ∈ sim (A), which implies the statement of the lemma. Suppose now i = t and write G = stm (F ) = At (m; n; k; r). For j ≥ r, set Fj = {A ∈ F : |A ∩ [n]| = j} and Gj = {A ∈ G : |A ∩ [n]| = j}. Clearly, for each A ∈ Fj , stm (A) ∈ Fj+1 subject to stm (A) ̸ = A, hence Gr ⊆ Fr . Note that

{ Gj =

α∪β :α ∈

(

[n]

)

j

with t ∈ α, β ∈

(

[n + 1, m] k−j

)}

,

j = r , . . . , k,

and n ≥ k + r and m − k > k − r. If Fk−1 ̸ ⊆ Gk−1 , that is, there is an A ∈ Fk−1 with t ̸ ∈ A, then we may find a B ∈ Gr with A ∩ B = ∅. Hence Fk−1 ⊆ Gk−1 , that is, Fk−1 is invariant under the action of stm , which yields Fk = Gk . From this it follows Fr = Gr , which implies Gr +1 ⊆ Fr +1 . This process is briefly expressed as Gr ⊆ Fr ⇒ Fk−1 ⊆ Gk−1 ⇒ Fk = Gk ⇒ Fr = Gr ⇒ Gr +1 ⊆ Fr +1 . Repeating this process we obtain that Fj = Gj for all r ≤ j ≤ k, that is, Fr = Gr . This completes the proof. □ Lemma 3.3. Let m, n, k and r be positive integers with m ≥ n ≥ k ≥ r > 0, n ≥ k + r and m − n ≥ k − r. If F is an intersecting subfamily of A(m, n, k, r), then

|F | ≤ f ∗ (m, n, k, r) =

)( ) k ( ∑ n−1 m−n i=r

i−1

k−i

,

and equality implies that F = At (m, n, k, r) for some t ∈ [n] unless n = 2k = 2r or m − n = k − r > 0.

(2)

J. Wang, H. Zhang / Discrete Mathematics 340 (2017) 140–144

143

Proof. Note that A(m, n, k, r) = k for k = r, and A(m, n, k, r) = k for m = n + k − r. In both cases, the classical Erdős–Ko–Rado theorem implies inequality (2), and if equality holds, then F is a t-star for some t ∈ [n] unless n = 2k = 2r or m − n = k − r > 0. Suppose k > r. We apply induction on m to prove inequality (2). We have seen that it holds for m = n + k − r. Assume that m > n + k − r. It is easily seen that A(m, n, k, r) is compressed. By Lemma 3.2 we may assume that sim (F ) = F for each i < m. We now consider the canonical decomposition of F = F0 ∪ F1 , where F0 = {A ∈ F : m ̸ ∈ A} and F1 = {A ∈ F : m ∈ A}. Clearly, F0 is an intersecting subfamily of A(m − 1, n, k, r) with m − 1 ≥ n + k − r, and F1′ = {A \ {m} : A ∈ F1 } is an intersecting subfamily of A(m − 1, n, k − 1, r) with m − 1 > n + k − 1 − r. Then, by the induction hypothesis we have

([n])

([m])

|F | = |F0 | + |F1 | ≤ f ∗ (m − 1, n, k, r) + f ∗ (m − 1, n, k − 1, r) = f ∗ (m, n, k, r). Suppose that the equality holds. Then |F0 | = f ∗ (m − 1, n, k, r) and |F1 | = f ∗ (m − 1, n, k − 1, r). Keeping m > n + k − r in mind, we apply induction on k to show F = At (m, n, k, r) for some t ∈ [n]. It is clearly true for k = r unless n = 2k = 2r. Assume that the result holds for k − 1 ≥ r. By the induction hypothesis we have that F1′ = At (m − 1, n, k − 1, r) for some t ∈ [n] because n ≥ k + r > k − 1 + r ≥ 2r. Note that F0 ∪ F1′ is intersecting as F0 and F1′ are intersecting. Clearly, for each set A in A(m; n; k; r) \ At (m; n; k; r), we can choose a set B in At (m − 1; n; k − 1; r) such that A ∩ B = ∅. Since F1′ = At (m − 1; n; k − 1; r), we therefore have F ⊆ At (m; n; k; r), and the equality follows from |F | = |At (m; n; k; r)|. □ 4. Proof of Theorem 1.1

([m])

If m = 2k, then for each A ∈ k , either A ∈ Ht (m; n; k; r) or [m] \ A ∈ Ht (m; n; k; r) for t ∈ [m], so Ht (m; n; k; r) is ([m]) a maximum intersecting subfamily of k . Assume now m > 2k. Let F be a maximum (m, n, k, r)-intersecting family. We ∗ have seen that F = A(m, n, k, r) ∪ F , where F ∗ is a maximum intersecting subfamily of B(m, n, k, r). Let us consider the case n = 2r − 2:

) ( )} [2r − 2] [2r − 1, m] B(m, 2r − 2, k, r) = A ∪ B : A ∈ ,B ∈ . r −1 k−r +1 ([2r −1,m]) Since m > 2k, |[2r − 1, m]| > 2(k − r + 1) so that k−r +1 is not an intersecting family. From this it follows that F ∗ = ([2r −2]) BT (m, 2r − 2, k, r), that is, F = HT (m, 2r − 2, k, r), for some maximum intersecting family T in r −1 (cf. [5,8,9], or [7, {

(

Proof of Theorem 10]). Assume n < 2r − 2. By Lemma 3.2 we may assume that F is invariant under the action of sim for each i < m, and consider the canonical decomposition F0 = {A ∈ F : m ̸ ∈ A}, F1 = {A ∈ F : m ∈ A} and F1′ = {A \ {m} : A ∈ F1 }. Then F0 is an (m − 1, n, k, r)-intersecting family, and F1′ is an (m − 1, n, k − 1, r)-intersecting family if k > r. The proof will be completed by induction on m. We have seen that the result is true for m = 2k. Assume now m > 2k. There are two cases to be considered. First, suppose k = r. In this case, by the induction hypothesis, |F0 | ≤ h(m − 1, n, k, k) and equality holds if and only if F0 = Ht1 (m − 1, n, k, k) for some t1 ∈ [n], except for m − 1 = 2k; and F1′ ⊆ A(m − 1, n, k − 1, n − k + 1). It is easy to check that the parameters satisfy the conditions in Lemma 3.3, so |F1′ | ≤ f ∗ (m − 1, n, k − 1, n − k + 1), and equality implies F1′ = At2 (m − 1, n, k − 1, n − k + 1) for some t2 ∈ [n]. By Lemmas 3.3 and 3.1(3) we have that

|F | = |F0 | + |F1 | ≤ h(m − 1, n, k, k) + f ∗ (m − 1, n, k − 1, n − k + 1) = h(m, n, k, k). Then maximality of F implies the equality. In this case it is easy to show t1 = t2 , hence F = Ht (m, n, k, k) for some t ∈ [n]. Next, suppose k > r. In this case, we have seen that F0 is an (m − 1, n, k, r)-intersecting family, and F1′ is an (m − 1, n, k − 1, r)-intersecting family. By the induction hypothesis and Lemma 3.1(4) we have that

|F | = |F0 | + |F1 | ≤ h(m − 1, n, k, r) + h(m − 1, n, k − 1, r) = h(m, n, k, r). Similarly to the first case, the maximality of F implies F = Ht (m, n, k, r) for some t ∈ [n]. □ Acknowledgments The first author is supported by the National Natural Science Foundation of China (Nos. 11171224 and 11231004); the second author is supported by the National Natural Science Foundation of China (No. 11371327). References [1] P. Borg, Intersecting families of sets and permutations: a survey, in: A.R. Baswell (Ed.), in: Advances in Mathematics Research, vol. 16, Nova Science Publishers, 2011, pp. 283–299. [2] M. Deza, P. Frankl, Erdős–Ko–Rado theorem–22 years later, SIAM J. Algebr. Discrete Methods 4 (1983) 419–431.

144 [3] [4] [5] [6] [7] [8] [9]

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