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Intersecting families, cross-intersecting families, and a proof of a conjecture of Feghali, Johnson and Thomas
Discrete Mathematics 341 (2018) 1331–1335
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Intersecting families, cross-intersecting families, and a proof of a conjecture of Feghali, Johnson and Thomas Peter Borg Department of Mathematics, Faculty of Science, University of Malta, Malta
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Article history: Received 12 April 2017 Received in revised form 31 January 2018 Accepted 1 February 2018 Available online 2 March 2018 Keywords: Intersecting family Cross-intersecting families Erdős–Ko–Rado theorem Feghali–Johnson–Thomas conjecture
a b s t r a c t A family A of sets is said to be intersecting if every two sets in A intersect. Two families A and B are said to be cross-intersecting if each set in A intersects each set in B. For a positive integer n, let [n] = {1, . . . , n} and Sn = {A ⊆ [n] : 1 ∈ A}. We extend the Erdős–Ko–Rado Theorem by showing that if A and B are non-empty cross-intersecting families of subsets of [n], A is intersecting, and a0 , a1 , . . . , an , b0 , b1 , . . . , bn are non-negative real numbers such that ai + bi ≥ an−i + bn−i and an−i ≥ bi for each i ≤ n/2, then
∑ A ∈A
a|A| +
∑ B∈B
b|B| ≤
∑ A∈Sn
a|A| +
∑
b|B| .
B∈Sn
For a graph G and an integer r ≥ 1, let IG (r) denote the family of r-element independent sets of G. Inspired by a problem of Holroyd and Talbot, Feghali, Johnson and Thomas conjectured that if r < n and G is a depth-two claw with n leaves, then G has a vertex v such that {A ∈ IG (r) : v ∈ A} is a largest intersecting subfamily of IG (r) . They proved this for r ≤ n+2 1 . We use the result above to prove the full conjecture. © 2018 Elsevier B.V. All rights reserved.
1. Introduction Unless otherwise stated, we shall use small letters such as x to denote non-negative integers or elements of a set, capital letters such as X to denote sets, and calligraphic letters such as F to denote families (that is, sets whose members are sets themselves). It is to be assumed that arbitrary sets and families are finite. We call a set A an r-element set if its size |A| is r, that is, if it contains exactly r elements (also called members). The set {1, 2, . . .} of positive integers is denoted by N. For any integer n ≥ 0, the set {i ∈ N : i ≤ n} is denoted by [n]. Note that [0] is the empty set by 2X . The family of r-element ( X )∅. For a set X , the power set of X (that is, {A : A ⊆ X }) is denoted (r) subsets of X is denoted by r . The family of r-element sets in a family F is denoted by F . If F ⊆ 2X and x ∈ X , then the family {F ∈ F : x ∈ F } is denoted by F (x) and called a star of F . We say that a set A intersects a set B if A and B have at least one common element ⋃(that is, A ∩ B ̸= ∅). A family A is said to be intersecting if for every A, B ∈ A, A and B intersect. The stars of a family F (with F ∈F F ̸ = ∅) are the simplest intersecting subfamilies of F . We say that F has the star property if at least one of the largest intersecting subfamilies of F is a star of F . One of the most popular endeavors in extremal set theory is that of determining the size of a largest intersecting subfamily of a given family F . This started in [18], which features the following classical result, known as the Erdős–Ko–Rado (EKR) Theorem. Theorem 1.1 (EKR Theorem [18]). If r ≤ n/2 and A is an intersecting subfamily of E-mail address: [email protected]. https://doi.org/10.1016/j.disc.2018.02.004 0012-365X/© 2018 Elsevier B.V. All rights reserved.
( [n ] ) r
, then |A| ≤
( n−1 ) r −1
.
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P. Borg / Discrete Mathematics 341 (2018) 1331–1335
( [n ] )
This means that r has the star property. There are various proofs of the EKR Theorem (see [14,29,31]), two of which are particularly short and beautiful: Katona’s [31], which introduced the elegant cycle method, and Daykin’s [14], using the fundamental Kruskal–Katona Theorem [30,32]. The EKR Theorem gave rise to some of the highlights in extremal set theory [1,20,29,34] and inspired many results that establish how large a system of sets can be under certain intersection conditions; see [6,15,21,22,23,26,27]. If A and B are families such that each set in A intersects each set in B, then A and B are said to be cross-intersecting. For intersecting subfamilies of a given family F , the natural question to ask is how large they can be. A natural variant of this intersection problem is the problem of maximizing the sum or the product of sizes of cross-intersecting subfamilies (not necessarily distinct or non-empty) of F . This has recently attracted much attention. The relation between the original intersection problem, the sum problem and the product problem is studied in [7]. Solutions have been obtained for various families; most of the known results are referenced in [8,9], which treat the product problem for families of subsets of [n] of size at most r. Here we consider the sum problem for the case where at least one of two cross-intersecting families A and B of subsets of [n] is an intersecting family. We actually consider a more general setting∑of weighted∑ sets, where each set ∑ of size i is assigned two non-negative integers ai and bi , and the objective is to maximize a + b . Note that | A | | B | A∈A B∈B A∈A a|A| = |A| if a0 = a1 = · · · = an = 1. Let Sn denote the star {A ⊆ [n] : 1 ∈ A} of 2[n] . In Section 2, we prove the following extension of the EKR Theorem. Theorem 1.2. If A and B are non-empty cross-intersecting families of subsets of [n], A is intersecting, and a0 , a1 , . . . , an , b0 , b1 , . . . , bn are non-negative real numbers such that ai + bi ≥ an−i + bn−i and an−i ≥ bi for each i ≤ n/2, then
∑
a| A | +
A∈A
∑
b|B| ≤
B∈B
∑
a| A | +
A∈Sn
∑
b|B| .
B∈Sn
The EKR Theorem is obtained by taking r ≤ n/2, B = A ⊆ r , and bi = 0 = ai − 1 for each i ∈ {0} ∪ [n]. We use Theorem 1.2 to prove a conjecture of Feghali, Johnson and Thomas [19, Conjecture 2.1]. Before stating the conjecture, we need some further definitions and notation. (X ) A graph G is a pair (X , Y ), where X is a set, called the vertex set of G, and Y is a subset of 2 and is called the edge set of G. The vertex set of G and the edge set of G are denoted by V (G) and E(G), respectively. An element of V (G) is called a vertex of G, and an element of E(G) is called an edge of G. We may represent an edge {v, w} by vw . If vw is an edge of G, then we say that v is adjacent to w (in G). A subset I of V (G) is an independent set of G if {v, w} ∈ / E(G) for every v, w ∈ I. Let IG denote the family of independent sets of G. An independent set J of G is maximal if J ̸ ⊆ I for each independent set I of G such that I ̸ = J. The size of a smallest maximal independent set of G is denoted by µ(G). Holroyd and Talbot introduced the problem of determining whether IG (r) has the star property for a given graph G and an integer r ≥ 1. The Holroyd–Talbot (HT) Conjecture [27, Conjecture 7] claims that IG (r) has the star property if µ(G) ≥ 2r. The author [4] proved that the conjecture is true if µ(G) is sufficiently large depending on r (see also [11, Lemma 4.4 and Theorem 1.4]). By the EKR Theorem, the conjecture is true if G has no edges. The HT Conjecture has been verified for several classes of graphs [12,13,24–28,33,35]. As demonstrated in [13], for r > µ(G)/2, whether IG (r) has the star property or not depends on G and r (both cases are possible). A depth-two claw is a graph consisting of n pairwise disjoint edges x1 y1 , . . . , xn yn together with a vertex x0 ̸ ∈ {x1 , . . . , xn , y1 , . . . , yn } that is adjacent to each of y1 , . . . , yn . This graph will be denoted by Tn . Thus, Tn = ({x0 , x1 , . . . , xn , y1 , . . . , yn }, {x0 y1 , . . . , x0 yn , x1 y1 , . . . , xn yn }). For each i ∈ [n], we may take xi and yi to be (i, 1) and (i, 2), respectively. Let Xn = {xi : i ∈ [n]} and
( [n ] )
{ Ln =
{(i1 , j1 ), . . . , (ir , jr )} : r ∈ [n], {i1 , . . . , ir } ∈
(
[n] r
)
} , j1 , . . . , jr ∈ {1, 2} .
Note that ITn (r) = Ln (r) ∪
{
(
A ∪ {x0 } : A ∈
Xn r −1
)}
.
(1)
The family ITn (r) is empty for r > n + 1, and consists only of the set {x0 , x1 , . . . , xn } for r = n + 1. In [19], Feghali, Johnson and Thomas showed that ITn (r) does not have the star property for r = n, and they made the following conjecture. Conjecture 1.3 ([19]). If 1 ≤ r ≤ n − 1, then ITn (r) has the star property. They proved the conjecture for r ≤ Theorem 1.4 ([19]). If 1 ≤ r ≤
n+1 . 2
n+1 , 2
then ITn (r) has the star property.
In the next section, we settle the full conjecture, using Theorem 1.2 for r >
n+1 . 2
P. Borg / Discrete Mathematics 341 (2018) 1331–1335
1333
Theorem 1.5. Conjecture 1.3 is true. Moreover, for 1 ≤ r ≤ n − 1, ITn (r) (x1 ) is a largest intersecting subfamily of ITn (r) . Our proof for r ≤ 2n + 1 provides an alternative proof of Theorem 1.4. It is worth mentioning that the paper of Feghali, Johnson and Thomas [19] was inspired by [28], in which Hurlbert and Kamat tackled the HT problem for chordal graphs and trees, and conjectured that, for any tree T and any r ≥ 1, T has a leaf x such that IT (r) (x) is a largest star of IT (r) . It is easy to check that the Hurlbert–Kamat (HK) Conjecture holds for the tree Tn ; Theorem 1.5 is a stronger fact for r ≤ n − 1. Hurlbert and Kamat [28] proved their conjecture for r ≤ 4. Remarkably, the HK conjecture is false for r ≥ 5, as was independently shown by Baber [2], the author [10], and Feghali, Johnson and Thomas [19]. In [10], it is actually established that, for any k ≥ 3, there exists a tree T that has a vertex x such that x is not a leaf of T , |IT (r) (z)| < |IT (r) (x)| for any leaf z of T and any integer r with 5 ≤ r ≤ 2k + 1, and 2k + 1 is the largest integer s for which IT (s) (x) is non-empty. 2. Proofs In this section, we prove Theorems 1.2 and 1.5. Proof of Theorem 1.2. For each i ∈ {0} ∪ [n], let ci = |A(i) |ai + |B(i) |bi . Since A and B are non-empty and cross-intersecting, we have ∅ ̸ ∈ A and ∅ ̸ ∈ B, so A(0) = B(0) = ∅. Thus, c0 = 0. The result is immediate if n = 1. Suppose n > 1. Consider any positive integer r ≤ n/2. Since A and B are cross-intersecting and A is intersecting, [n] \ A ̸ ∈ A(n−r) for each A ∈ A(r) ∪ B(r) , so
|A(n−r) | ≤
(
n
)
n−r
− |{[n] \ A : A ∈ A(r) ∪ B(r) }| =
Similarly, since A and B are cross-intersecting, |B(n−r) | ≤
(n) r
(n) r
− |A(r) ∪ B(r) | =
(n) r
− |A(r) | − |B(r) \ A(r) |.
− |A(r) |. Thus,
) (( n ) ) − |A(r) | − |B(r) \ A(r) | an−r + − |A(r) | bn−r r r (n) = |A(r) |(ar + br − an−r − bn−r ) − |B(r) \ A(r) |(an−r − br ) + (an−r + bn−r ) r ( ) (n) n−1 ≤ (ar + br − an−r − bn−r ) + (an−r + bn−r ) r −1 r ( n−1 ) by Theorem 1.1 and the given conditions ar + br ≥ an−r + bn−r and an−r ≥ br . Therefore, cr + cn−r ≤ r −1 (ar + ( n−1 ) ( n−1 ) ( n−1 ) br ) + r (an−r + bn−r ) = r −1 (ar + br ) + n−r −1 (an−r + bn−r ). Note that if r = n/2, then we have cn/2 + cn/2 ≤ ( ) ( ) ( ) n−1 n−1 n−1 (a + b ) + (a + b ), and hence c ≤ (an/2 + bn/2 ). n / 2 n / 2 n / 2 n / 2 n / 2 n/2−1 n/2−1 n/2−1 cr + cn−r ≤ |A(r) |ar + |A(r) | + |B(r) \ A(r) | br +
(
)
(( n )
If n is odd, then n−1
n−1
∑
a| A | +
A∈A
∑
b|B| =
B∈B
2 ∑
(i)
(n−i)
(i)
(|A |ai + |A
|an−i + |B |bi + |B
(n−i)
|bn−i ) = c0 + cn +
i=0
2 ∑
(ci + cn−i )
i=1 n −1
≤ an + b n +
) 2 (( ∑ n−1 i−1
i=1
( (ai + bi ) +
n−1 n−i−1
)
) (an−i + bn−i )
=
∑
a| A | +
A∈Sn
∑
b|B| .
B∈Sn
Similarly, if n is even, then n
∑
a| A | +
A∈A
∑
b|B| = c0 + cn + cn/2 +
B∈B
−1 2 ∑
(ci + cn−i )
i=1 n
( ≤ an + bn + =
∑ A∈Sn
a| A | +
n−1 n/2 − 1
∑
) (an/2 + bn/2 ) +
−1 (( ) 2 ∑ n−1 i=1
i−1
( (ai + bi ) +
n−1 n−i−1
)
) (an−i + bn−i )
b|B| ,
B∈Sn
as required. □ In the proof of Theorem 1.5, we shall make use of established facts regarding certain compression operations. For any edge uv of a graph G, let γu,v : IG → IG be defined by
γu,v (A) =
(A \ {v}) ∪ {u} if v ∈ A, u ̸ ∈ A and (A \ {v}) ∪ {u} ∈ IG ;
{
A
otherwise,
1334
P. Borg / Discrete Mathematics 341 (2018) 1331–1335
and let Γu,v : 2IG → 2IG be the compression operation defined by
Γu,v (A) = {γu,v (A) : A ∈ A} ∪ {A ∈ A : γu,v (A) ∈ A}. For any x ∈ V (G), let NG (x) denote the set {y ∈ V (G) : xy ∈ E(G)}. The following is given by [13, Lemma 2.1] (which is actually stated for IG (r) but proved for IG ) and essentially originated in [26]. Lemma 2.1 ([13,26]). If G is a graph, uv ∈ E(G), E is an intersecting subfamily of IG , G = Γu,v (E ), G0 = {A ∈ G : v ̸ ∈ A}, G1 = {A ∈ G : v ∈ A} and G1′ = {A \ {v} : A ∈ G1 }, then: (i) G0 is intersecting; (ii) if |NG (u) \ ({v} ∪ NG (v ))| ≤ 1, then G1′ is intersecting; (iii) if NG (u) \ ({v} ∪ NG (v )) = ∅, then G0 ∪ G1′ is intersecting. For i ∈ [n], let δi : Ln → Ln be defined by
δi (A) = and let ∆i : 2
(A \ {(i, 2)}) ∪ {(i, 1)} if (i, 2) ∈ A;
{
Ln
A
otherwise, Ln
→2
be the compression operation defined by
∆i (A) = {δi (A) : A ∈ A} ∪ {A ∈ A : δi (A) ∈ A}. The following is a special case of [5, Corollary 3.2]. Lemma 2.2 ([5]). If E is an intersecting subfamily of Ln , and E ∗ = ∆n ◦ · · · ◦ ∆1 (E ), then |E ∩ F ∩ Xn | ≥ 1 for any E , F ∈ E ∗ . It is well known, and easy to prove, that compressions like Γu,v and ∆i preserve the sizes of the families they act on (see [21]). Proof of Theorem 1.5. The result is trivial for r = 1, so consider 2 ≤ r ≤ n − 1. Let R = ITn (r) . Let E be an intersecting subfamily of R. Let F = {A ∈ R : x1 ∈ A}. For any H ⊆ R,(let H ) 0 = {H ∈ H : x0 ̸∈ H }, H1 = {H ∈ H : x0 ∈ H } X and H1′ = {H \ {x0 } : H ∈ H1 }; by (1), H0 ⊆ Ln (r) and H1′ ⊆ r −n1 . It is well known that Ln (r) has the star property [15, Theorem 5.2] (see also [3,16,17]). Case 1: r ≤ 2n + 1. Note that, for A ∈ R, γy1 ,x0 (A) ̸ = A if and only if x0 ∈ A and x1 , y1 ̸ ∈ A. Let G = Γy1 ,x0 (E ). Then G ⊆ R (X ) and |G | = |E |. By parts (i) and (ii) of Lemma 2.1, G0 and G1′ are intersecting. Since r − 1 ≤ 2n and G1′ ⊆ r −n1 , Theorem 1.1
(
X
)
gives us |G1′ | ≤ |{A ∈ r −n1 : x1 ∈ A}| = |F1′ |. Since Ln (r) has the star property, |G0 | ≤ |{A ∈ Ln (r) : x1 ∈ A}| = |F0 |. Therefore, we have |E | = |G | = |G0 | + |G1′ | ≤ |F0 | + |F1′ | = |F |. Case 2: r ≥ 2n + 1. If E0 = ∅, then |E | = |E1′ | ≤ r −1 = |{A ∈ n r 0 : x1 ∈ A}| < |F |. If E1′ = ∅, then E = E0 ⊆ Ln (r) , and hence, since Ln (r) has the star property, |E |≤ |{A ∈ Ln (r) : x1 ∈ A}| < |F |. Now suppose E0 ̸ = ∅ and E1′ ̸ = ∅. Let E0 ∗ = ∆n ◦ · · · ◦ ∆1 (E0 ). Then E0 ∗ ⊆ Ln (r) . By Lemma 2.2,
(
n
(X
)
∪{x }
)
E ∩ F ∩ Xn ̸ = ∅ for any E , F ∈ E0 ∗ .
(2)
For any E ∈ E0 and any F ∈ E1 , we have ∅ ̸ = E ∩ F ⊆ Xn and E ∩ Xn ⊆ δi (E) for any i ∈ [n]. It clearly follows that E ∩ F ∩ Xn ̸ = ∅ for any E ∈ E0 ∗ and any F ∈ E1′ .
(3)
Let A = {E ∩ Xn : E ∈ E0 } and B (= E1). Then A ̸ = ∅ (as |E0 | = |E0 | > 0) and B ̸ = ∅. By (2), A is intersecting. By (3), A and n−i B are cross-intersecting. Let ai = r −i for each i ∈ {0} ∪ [r ], and let ai = 0 for each i ∈ [n] \ [r ]. Since r ≤ n − 1, we have ′
∗
∗
a0 > · · · > ar > ar +1 = · · · = an = 0.
(4)
Let br −1 = 1 and let bi = 0 for each i ∈ ({0} ∪ [n]) \ {r − 1}. Consider any i ≤ 2n . Since r ≥ 2n + 1, i ≤ r − 1. Suppose i < r − 1. Then bi = 0 ≤ an−i . If i < n/2, then, by (4), we have ai ≥ an−i + 1 (as i < 2n < r and a0 , . . . , an are integers), so ai + bi ≥ an−i + bn−i . If i = n/2, then n − i = i, so ai + bi = an−i + bn−i . Now suppose i = r − 1. Since i ≤ n/2 ≤ r − 1 = i, i = n/2 = r − 1. Thus, bn−i = bi = 1 and an−i = ai ≥ 1, and hence ai + bi = an−i + bn−i and an−i ≥ bi . We have shown that ai + bi ≥ an−i + bn−i and an−i ≥ bi for each i ≤ n/2. By Theorem 1.2,
∑ A∈A
a| A | +
∑
b|B| ≤
B∈B
∑
a| A | +
A∈Xn
∑
b|B| ,
B∈Xn
where Xn = {A ⊆ Xn : x1 ∈ A}. Now
|E0 | = |E0 ∗ | =
r r r ∑ ∑ ⏐{ ⏐{ }⏐ ∑ }⏐ ∑ ⏐ E ∈ E0 ∗ : E ∩ Xn ∈ A(i) ⏐ ≤ ⏐ E ∈ Ln (r) : E ∩ Xn ∈ A(i) ⏐ = |A(i) |ai = a| A | i=0
i=0
i=0
A∈A
P. Borg / Discrete Mathematics 341 (2018) 1331–1335
and |E1 | = |B| =
∑
B∈B br −1
|E | = |E0 | + |E1 | ≤
= ∑
∑
B∈B b|B| .
a| A | +
A∈A
∑ B∈B
Clearly, |F0 | =
b|B| ≤
∑ A∈Xn
∑
a| A | +
A∈Xn a|A|
∑
and |F1 | =
∑
B∈Xn b|B| .
1335
We have
b|B| = |F0 | + |F1 | = |F |,
B∈Xn
and hence the result is proved. □ Remark 2.3. Note that if we assume Theorem 1.4, then we only need Case 2 of the proof of Theorem 1.5. Also note that the case r = 2n + 1 is proved in both Case 1 and Case 2. Thus, Case 1 allows us to restrict Case 2 to r > 2n + 1, in which case we do not have i = r − 1 when considering i ≤ 2n (as 2n < r − 1). Acknowledgments The author wishes to thank the anonymous referees for helpful remarks that led to an improvement in the presentation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
R. Ahlswede, L.H. Khachatrian, The complete intersection theorem for systems of finite sets, European J. Combin. 18 (1997) 125–136. R. Baber, Some Results in Extremal Combinatorics (Ph.D. thesis), University College London, 2011. B. Bollobás, I. Leader, An Erdős–Ko–Rado theorem for signed sets, Comput. Math. Appl. 34 (1997) 9–13. P. Borg, Extremal t-intersecting sub-families of hereditary families, J. Lond. Math. Soc. 79 (2009) 167–185. P. Borg, On t-intersecting families of signed sets and permutations, Discrete Math. 309 (2009) 3310–3317. P. Borg, Intersecting families of sets and permutations: a survey, in: A.R. Baswell (Ed.), Advances in Mathematics Research, Vol. 16, Nova Science Publishers, Inc., 2011, pp. 283–299. P. Borg, The maximum sum and the maximum product of sizes of cross-intersecting families, European J. Combin. 35 (2014) 117–130. P. Borg, A cross-intersection theorem for subsets of a set, Bull. Lond. Math. Soc. 47 (2015) 248–256. P. Borg, The maximum product of weights of cross-intersecting families, J. Lond. Math. Soc. 94 (2016) 993–1018. P. Borg, Stars on trees, Discrete Math. 340 (2017) 1046–1049. P. Borg, The maximum product of sizes of cross-intersecting families, Discrete Math. 340 (2017) 2307–2317. P. Borg, F. Holroyd, The Erdős–Ko–Rado properties of set systems defined by double partitions, Discrete Math. 309 (2009) 4754–4761. P. Borg, F. Holroyd, The Erdős–Ko–Rado property of various graphs containing singletons, Discrete Math. 309 (2009) 2877–2885. D.E. Daykin, Erdős–Ko–Rado from Kruskal–Katona, J. Combin. Theory Ser. A 17 (1974) 254–255. M. Deza, P. Frankl, The Erdős–Ko–Rado theorem—22 years later, SIAM J. Algebr. Discrete Methods 4 (1983) 419–431. K. Engel, An Erdős–Ko–Rado theorem for the subcubes of a cube, Combinatorica 4 (1984) 133–140. P.L. Erdős, U. Faigle, W. Kern, A group-theoretic setting for some intersecting Sperner families, Combin. Probab. Comput. 1 (1992) 323–334. P. Erdős, C. Ko, R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. (2) 12 (1961) 313–320. C. Feghali, M. Johnson, D. Thomas, Erdős–Ko–Rado theorems for a family of trees, Discrete Appl. Math. 236 (2018) 464–471. P. Frankl, The Erdős-Ko-Rado Theorem is true for n = ckt, in: Proc. Fifth Hung. Comb. Coll., North-Holland, Amsterdam, 1978, pp. 365–375. P. Frankl, The shifting technique in extremal set theory, in: C. Whitehead (Ed.), Surveys in Combinatorics, Cambridge Univ. Press, London/New York, 1987, pp. 81–110. P. Frankl, Extremal set systems, in: R.L. Graham, M. Grötschel, L. Lovász (Eds.), Handbook of Combinatorics, Vol. 2, Elsevier, Amsterdam, 1995, pp. 1293–1329. P. Frankl, N. Tokushige, Invitation to intersection problems for finite sets, J. Combin. Theory Ser. A 144 (2016) 157–211. A.J.W. Hilton, F.C. Holroyd, C.L. Spencer, King Arthur and his knights with two round tables, Q. J. Math. 62 (2011) 625–635. A.J.W. Hilton, C.L. Spencer, A generalization of Talbot’s theorem about King Arthur and his Knights of the Round Table, J. Combin. Theory Ser. A 116 (2009) 1023–1033. F. Holroyd, C. Spencer, J. Talbot, Compression and Erdős–Ko–Rado graphs, Discrete Math. 293 (2005) 155–164. F. Holroyd, J. Talbot, Graphs with the Erdős–Ko–Rado property, Discrete Math. 293 (2005) 165–176. G. Hurlbert, V. Kamat, Erdős–Ko–Rado theorems for chordal graphs and trees, J. Combin. Theory Ser. A 118 (2011) 829–841. G.O.H. Katona, Intersection theorems for systems of finite sets, Acta Math. Acad. Sci. Hungar. 15 (1964) 329–337. G.O.H. Katona, A theorem of finite sets, in: Theory of Graphs, in: Proc. Colloq. Tihany, Akadémiai Kiadó, 1968, pp. 187–207. G.O.H. Katona, A simple proof of the Erdős–Chao Ko–Rado theorem, J. Combin. Theory Ser. B 13 (1972) 183–184. J.B. Kruskal, The number of simplices in a complex, Mathematical Optimization Techniques, University of California Press, Berkeley, California, 1963, pp. 251–278. J. Talbot, Intersecting families of separated sets, J. Lond. Math. Soc. 68 (2003) 37–51. R.M. Wilson, The exact bound in the Erdős–Ko–Rado theorem, Combinatorica 4 (1984) 247–257. R. Woodroofe, Erdős–Ko–Rado theorems for simplicial complexes, J. Combin. Theory Ser. A 118 (2011) 1218–1227.
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Intersecting families, cross-intersecting families, and a proof of a conjecture of Feghali, Johnson and Thomas Peter Borg Department of Mathematics, Faculty of Science, University of Malta, Malta
article
info
Article history: Received 12 April 2017 Received in revised form 31 January 2018 Accepted 1 February 2018 Available online 2 March 2018 Keywords: Intersecting family Cross-intersecting families Erdős–Ko–Rado theorem Feghali–Johnson–Thomas conjecture
a b s t r a c t A family A of sets is said to be intersecting if every two sets in A intersect. Two families A and B are said to be cross-intersecting if each set in A intersects each set in B. For a positive integer n, let [n] = {1, . . . , n} and Sn = {A ⊆ [n] : 1 ∈ A}. We extend the Erdős–Ko–Rado Theorem by showing that if A and B are non-empty cross-intersecting families of subsets of [n], A is intersecting, and a0 , a1 , . . . , an , b0 , b1 , . . . , bn are non-negative real numbers such that ai + bi ≥ an−i + bn−i and an−i ≥ bi for each i ≤ n/2, then
∑ A ∈A
a|A| +
∑ B∈B
b|B| ≤
∑ A∈Sn
a|A| +
∑
b|B| .
B∈Sn
For a graph G and an integer r ≥ 1, let IG (r) denote the family of r-element independent sets of G. Inspired by a problem of Holroyd and Talbot, Feghali, Johnson and Thomas conjectured that if r < n and G is a depth-two claw with n leaves, then G has a vertex v such that {A ∈ IG (r) : v ∈ A} is a largest intersecting subfamily of IG (r) . They proved this for r ≤ n+2 1 . We use the result above to prove the full conjecture. © 2018 Elsevier B.V. All rights reserved.
1. Introduction Unless otherwise stated, we shall use small letters such as x to denote non-negative integers or elements of a set, capital letters such as X to denote sets, and calligraphic letters such as F to denote families (that is, sets whose members are sets themselves). It is to be assumed that arbitrary sets and families are finite. We call a set A an r-element set if its size |A| is r, that is, if it contains exactly r elements (also called members). The set {1, 2, . . .} of positive integers is denoted by N. For any integer n ≥ 0, the set {i ∈ N : i ≤ n} is denoted by [n]. Note that [0] is the empty set by 2X . The family of r-element ( X )∅. For a set X , the power set of X (that is, {A : A ⊆ X }) is denoted (r) subsets of X is denoted by r . The family of r-element sets in a family F is denoted by F . If F ⊆ 2X and x ∈ X , then the family {F ∈ F : x ∈ F } is denoted by F (x) and called a star of F . We say that a set A intersects a set B if A and B have at least one common element ⋃(that is, A ∩ B ̸= ∅). A family A is said to be intersecting if for every A, B ∈ A, A and B intersect. The stars of a family F (with F ∈F F ̸ = ∅) are the simplest intersecting subfamilies of F . We say that F has the star property if at least one of the largest intersecting subfamilies of F is a star of F . One of the most popular endeavors in extremal set theory is that of determining the size of a largest intersecting subfamily of a given family F . This started in [18], which features the following classical result, known as the Erdős–Ko–Rado (EKR) Theorem. Theorem 1.1 (EKR Theorem [18]). If r ≤ n/2 and A is an intersecting subfamily of E-mail address: [email protected]. https://doi.org/10.1016/j.disc.2018.02.004 0012-365X/© 2018 Elsevier B.V. All rights reserved.
( [n ] ) r
, then |A| ≤
( n−1 ) r −1
.
1332
P. Borg / Discrete Mathematics 341 (2018) 1331–1335
( [n ] )
This means that r has the star property. There are various proofs of the EKR Theorem (see [14,29,31]), two of which are particularly short and beautiful: Katona’s [31], which introduced the elegant cycle method, and Daykin’s [14], using the fundamental Kruskal–Katona Theorem [30,32]. The EKR Theorem gave rise to some of the highlights in extremal set theory [1,20,29,34] and inspired many results that establish how large a system of sets can be under certain intersection conditions; see [6,15,21,22,23,26,27]. If A and B are families such that each set in A intersects each set in B, then A and B are said to be cross-intersecting. For intersecting subfamilies of a given family F , the natural question to ask is how large they can be. A natural variant of this intersection problem is the problem of maximizing the sum or the product of sizes of cross-intersecting subfamilies (not necessarily distinct or non-empty) of F . This has recently attracted much attention. The relation between the original intersection problem, the sum problem and the product problem is studied in [7]. Solutions have been obtained for various families; most of the known results are referenced in [8,9], which treat the product problem for families of subsets of [n] of size at most r. Here we consider the sum problem for the case where at least one of two cross-intersecting families A and B of subsets of [n] is an intersecting family. We actually consider a more general setting∑of weighted∑ sets, where each set ∑ of size i is assigned two non-negative integers ai and bi , and the objective is to maximize a + b . Note that | A | | B | A∈A B∈B A∈A a|A| = |A| if a0 = a1 = · · · = an = 1. Let Sn denote the star {A ⊆ [n] : 1 ∈ A} of 2[n] . In Section 2, we prove the following extension of the EKR Theorem. Theorem 1.2. If A and B are non-empty cross-intersecting families of subsets of [n], A is intersecting, and a0 , a1 , . . . , an , b0 , b1 , . . . , bn are non-negative real numbers such that ai + bi ≥ an−i + bn−i and an−i ≥ bi for each i ≤ n/2, then
∑
a| A | +
A∈A
∑
b|B| ≤
B∈B
∑
a| A | +
A∈Sn
∑
b|B| .
B∈Sn
The EKR Theorem is obtained by taking r ≤ n/2, B = A ⊆ r , and bi = 0 = ai − 1 for each i ∈ {0} ∪ [n]. We use Theorem 1.2 to prove a conjecture of Feghali, Johnson and Thomas [19, Conjecture 2.1]. Before stating the conjecture, we need some further definitions and notation. (X ) A graph G is a pair (X , Y ), where X is a set, called the vertex set of G, and Y is a subset of 2 and is called the edge set of G. The vertex set of G and the edge set of G are denoted by V (G) and E(G), respectively. An element of V (G) is called a vertex of G, and an element of E(G) is called an edge of G. We may represent an edge {v, w} by vw . If vw is an edge of G, then we say that v is adjacent to w (in G). A subset I of V (G) is an independent set of G if {v, w} ∈ / E(G) for every v, w ∈ I. Let IG denote the family of independent sets of G. An independent set J of G is maximal if J ̸ ⊆ I for each independent set I of G such that I ̸ = J. The size of a smallest maximal independent set of G is denoted by µ(G). Holroyd and Talbot introduced the problem of determining whether IG (r) has the star property for a given graph G and an integer r ≥ 1. The Holroyd–Talbot (HT) Conjecture [27, Conjecture 7] claims that IG (r) has the star property if µ(G) ≥ 2r. The author [4] proved that the conjecture is true if µ(G) is sufficiently large depending on r (see also [11, Lemma 4.4 and Theorem 1.4]). By the EKR Theorem, the conjecture is true if G has no edges. The HT Conjecture has been verified for several classes of graphs [12,13,24–28,33,35]. As demonstrated in [13], for r > µ(G)/2, whether IG (r) has the star property or not depends on G and r (both cases are possible). A depth-two claw is a graph consisting of n pairwise disjoint edges x1 y1 , . . . , xn yn together with a vertex x0 ̸ ∈ {x1 , . . . , xn , y1 , . . . , yn } that is adjacent to each of y1 , . . . , yn . This graph will be denoted by Tn . Thus, Tn = ({x0 , x1 , . . . , xn , y1 , . . . , yn }, {x0 y1 , . . . , x0 yn , x1 y1 , . . . , xn yn }). For each i ∈ [n], we may take xi and yi to be (i, 1) and (i, 2), respectively. Let Xn = {xi : i ∈ [n]} and
( [n ] )
{ Ln =
{(i1 , j1 ), . . . , (ir , jr )} : r ∈ [n], {i1 , . . . , ir } ∈
(
[n] r
)
} , j1 , . . . , jr ∈ {1, 2} .
Note that ITn (r) = Ln (r) ∪
{
(
A ∪ {x0 } : A ∈
Xn r −1
)}
.
(1)
The family ITn (r) is empty for r > n + 1, and consists only of the set {x0 , x1 , . . . , xn } for r = n + 1. In [19], Feghali, Johnson and Thomas showed that ITn (r) does not have the star property for r = n, and they made the following conjecture. Conjecture 1.3 ([19]). If 1 ≤ r ≤ n − 1, then ITn (r) has the star property. They proved the conjecture for r ≤ Theorem 1.4 ([19]). If 1 ≤ r ≤
n+1 . 2
n+1 , 2
then ITn (r) has the star property.
In the next section, we settle the full conjecture, using Theorem 1.2 for r >
n+1 . 2
P. Borg / Discrete Mathematics 341 (2018) 1331–1335
1333
Theorem 1.5. Conjecture 1.3 is true. Moreover, for 1 ≤ r ≤ n − 1, ITn (r) (x1 ) is a largest intersecting subfamily of ITn (r) . Our proof for r ≤ 2n + 1 provides an alternative proof of Theorem 1.4. It is worth mentioning that the paper of Feghali, Johnson and Thomas [19] was inspired by [28], in which Hurlbert and Kamat tackled the HT problem for chordal graphs and trees, and conjectured that, for any tree T and any r ≥ 1, T has a leaf x such that IT (r) (x) is a largest star of IT (r) . It is easy to check that the Hurlbert–Kamat (HK) Conjecture holds for the tree Tn ; Theorem 1.5 is a stronger fact for r ≤ n − 1. Hurlbert and Kamat [28] proved their conjecture for r ≤ 4. Remarkably, the HK conjecture is false for r ≥ 5, as was independently shown by Baber [2], the author [10], and Feghali, Johnson and Thomas [19]. In [10], it is actually established that, for any k ≥ 3, there exists a tree T that has a vertex x such that x is not a leaf of T , |IT (r) (z)| < |IT (r) (x)| for any leaf z of T and any integer r with 5 ≤ r ≤ 2k + 1, and 2k + 1 is the largest integer s for which IT (s) (x) is non-empty. 2. Proofs In this section, we prove Theorems 1.2 and 1.5. Proof of Theorem 1.2. For each i ∈ {0} ∪ [n], let ci = |A(i) |ai + |B(i) |bi . Since A and B are non-empty and cross-intersecting, we have ∅ ̸ ∈ A and ∅ ̸ ∈ B, so A(0) = B(0) = ∅. Thus, c0 = 0. The result is immediate if n = 1. Suppose n > 1. Consider any positive integer r ≤ n/2. Since A and B are cross-intersecting and A is intersecting, [n] \ A ̸ ∈ A(n−r) for each A ∈ A(r) ∪ B(r) , so
|A(n−r) | ≤
(
n
)
n−r
− |{[n] \ A : A ∈ A(r) ∪ B(r) }| =
Similarly, since A and B are cross-intersecting, |B(n−r) | ≤
(n) r
(n) r
− |A(r) ∪ B(r) | =
(n) r
− |A(r) | − |B(r) \ A(r) |.
− |A(r) |. Thus,
) (( n ) ) − |A(r) | − |B(r) \ A(r) | an−r + − |A(r) | bn−r r r (n) = |A(r) |(ar + br − an−r − bn−r ) − |B(r) \ A(r) |(an−r − br ) + (an−r + bn−r ) r ( ) (n) n−1 ≤ (ar + br − an−r − bn−r ) + (an−r + bn−r ) r −1 r ( n−1 ) by Theorem 1.1 and the given conditions ar + br ≥ an−r + bn−r and an−r ≥ br . Therefore, cr + cn−r ≤ r −1 (ar + ( n−1 ) ( n−1 ) ( n−1 ) br ) + r (an−r + bn−r ) = r −1 (ar + br ) + n−r −1 (an−r + bn−r ). Note that if r = n/2, then we have cn/2 + cn/2 ≤ ( ) ( ) ( ) n−1 n−1 n−1 (a + b ) + (a + b ), and hence c ≤ (an/2 + bn/2 ). n / 2 n / 2 n / 2 n / 2 n / 2 n/2−1 n/2−1 n/2−1 cr + cn−r ≤ |A(r) |ar + |A(r) | + |B(r) \ A(r) | br +
(
)
(( n )
If n is odd, then n−1
n−1
∑
a| A | +
A∈A
∑
b|B| =
B∈B
2 ∑
(i)
(n−i)
(i)
(|A |ai + |A
|an−i + |B |bi + |B
(n−i)
|bn−i ) = c0 + cn +
i=0
2 ∑
(ci + cn−i )
i=1 n −1
≤ an + b n +
) 2 (( ∑ n−1 i−1
i=1
( (ai + bi ) +
n−1 n−i−1
)
) (an−i + bn−i )
=
∑
a| A | +
A∈Sn
∑
b|B| .
B∈Sn
Similarly, if n is even, then n
∑
a| A | +
A∈A
∑
b|B| = c0 + cn + cn/2 +
B∈B
−1 2 ∑
(ci + cn−i )
i=1 n
( ≤ an + bn + =
∑ A∈Sn
a| A | +
n−1 n/2 − 1
∑
) (an/2 + bn/2 ) +
−1 (( ) 2 ∑ n−1 i=1
i−1
( (ai + bi ) +
n−1 n−i−1
)
) (an−i + bn−i )
b|B| ,
B∈Sn
as required. □ In the proof of Theorem 1.5, we shall make use of established facts regarding certain compression operations. For any edge uv of a graph G, let γu,v : IG → IG be defined by
γu,v (A) =
(A \ {v}) ∪ {u} if v ∈ A, u ̸ ∈ A and (A \ {v}) ∪ {u} ∈ IG ;
{
A
otherwise,
1334
P. Borg / Discrete Mathematics 341 (2018) 1331–1335
and let Γu,v : 2IG → 2IG be the compression operation defined by
Γu,v (A) = {γu,v (A) : A ∈ A} ∪ {A ∈ A : γu,v (A) ∈ A}. For any x ∈ V (G), let NG (x) denote the set {y ∈ V (G) : xy ∈ E(G)}. The following is given by [13, Lemma 2.1] (which is actually stated for IG (r) but proved for IG ) and essentially originated in [26]. Lemma 2.1 ([13,26]). If G is a graph, uv ∈ E(G), E is an intersecting subfamily of IG , G = Γu,v (E ), G0 = {A ∈ G : v ̸ ∈ A}, G1 = {A ∈ G : v ∈ A} and G1′ = {A \ {v} : A ∈ G1 }, then: (i) G0 is intersecting; (ii) if |NG (u) \ ({v} ∪ NG (v ))| ≤ 1, then G1′ is intersecting; (iii) if NG (u) \ ({v} ∪ NG (v )) = ∅, then G0 ∪ G1′ is intersecting. For i ∈ [n], let δi : Ln → Ln be defined by
δi (A) = and let ∆i : 2
(A \ {(i, 2)}) ∪ {(i, 1)} if (i, 2) ∈ A;
{
Ln
A
otherwise, Ln
→2
be the compression operation defined by
∆i (A) = {δi (A) : A ∈ A} ∪ {A ∈ A : δi (A) ∈ A}. The following is a special case of [5, Corollary 3.2]. Lemma 2.2 ([5]). If E is an intersecting subfamily of Ln , and E ∗ = ∆n ◦ · · · ◦ ∆1 (E ), then |E ∩ F ∩ Xn | ≥ 1 for any E , F ∈ E ∗ . It is well known, and easy to prove, that compressions like Γu,v and ∆i preserve the sizes of the families they act on (see [21]). Proof of Theorem 1.5. The result is trivial for r = 1, so consider 2 ≤ r ≤ n − 1. Let R = ITn (r) . Let E be an intersecting subfamily of R. Let F = {A ∈ R : x1 ∈ A}. For any H ⊆ R,(let H ) 0 = {H ∈ H : x0 ̸∈ H }, H1 = {H ∈ H : x0 ∈ H } X and H1′ = {H \ {x0 } : H ∈ H1 }; by (1), H0 ⊆ Ln (r) and H1′ ⊆ r −n1 . It is well known that Ln (r) has the star property [15, Theorem 5.2] (see also [3,16,17]). Case 1: r ≤ 2n + 1. Note that, for A ∈ R, γy1 ,x0 (A) ̸ = A if and only if x0 ∈ A and x1 , y1 ̸ ∈ A. Let G = Γy1 ,x0 (E ). Then G ⊆ R (X ) and |G | = |E |. By parts (i) and (ii) of Lemma 2.1, G0 and G1′ are intersecting. Since r − 1 ≤ 2n and G1′ ⊆ r −n1 , Theorem 1.1
(
X
)
gives us |G1′ | ≤ |{A ∈ r −n1 : x1 ∈ A}| = |F1′ |. Since Ln (r) has the star property, |G0 | ≤ |{A ∈ Ln (r) : x1 ∈ A}| = |F0 |. Therefore, we have |E | = |G | = |G0 | + |G1′ | ≤ |F0 | + |F1′ | = |F |. Case 2: r ≥ 2n + 1. If E0 = ∅, then |E | = |E1′ | ≤ r −1 = |{A ∈ n r 0 : x1 ∈ A}| < |F |. If E1′ = ∅, then E = E0 ⊆ Ln (r) , and hence, since Ln (r) has the star property, |E |≤ |{A ∈ Ln (r) : x1 ∈ A}| < |F |. Now suppose E0 ̸ = ∅ and E1′ ̸ = ∅. Let E0 ∗ = ∆n ◦ · · · ◦ ∆1 (E0 ). Then E0 ∗ ⊆ Ln (r) . By Lemma 2.2,
(
n
(X
)
∪{x }
)
E ∩ F ∩ Xn ̸ = ∅ for any E , F ∈ E0 ∗ .
(2)
For any E ∈ E0 and any F ∈ E1 , we have ∅ ̸ = E ∩ F ⊆ Xn and E ∩ Xn ⊆ δi (E) for any i ∈ [n]. It clearly follows that E ∩ F ∩ Xn ̸ = ∅ for any E ∈ E0 ∗ and any F ∈ E1′ .
(3)
Let A = {E ∩ Xn : E ∈ E0 } and B (= E1). Then A ̸ = ∅ (as |E0 | = |E0 | > 0) and B ̸ = ∅. By (2), A is intersecting. By (3), A and n−i B are cross-intersecting. Let ai = r −i for each i ∈ {0} ∪ [r ], and let ai = 0 for each i ∈ [n] \ [r ]. Since r ≤ n − 1, we have ′
∗
∗
a0 > · · · > ar > ar +1 = · · · = an = 0.
(4)
Let br −1 = 1 and let bi = 0 for each i ∈ ({0} ∪ [n]) \ {r − 1}. Consider any i ≤ 2n . Since r ≥ 2n + 1, i ≤ r − 1. Suppose i < r − 1. Then bi = 0 ≤ an−i . If i < n/2, then, by (4), we have ai ≥ an−i + 1 (as i < 2n < r and a0 , . . . , an are integers), so ai + bi ≥ an−i + bn−i . If i = n/2, then n − i = i, so ai + bi = an−i + bn−i . Now suppose i = r − 1. Since i ≤ n/2 ≤ r − 1 = i, i = n/2 = r − 1. Thus, bn−i = bi = 1 and an−i = ai ≥ 1, and hence ai + bi = an−i + bn−i and an−i ≥ bi . We have shown that ai + bi ≥ an−i + bn−i and an−i ≥ bi for each i ≤ n/2. By Theorem 1.2,
∑ A∈A
a| A | +
∑
b|B| ≤
B∈B
∑
a| A | +
A∈Xn
∑
b|B| ,
B∈Xn
where Xn = {A ⊆ Xn : x1 ∈ A}. Now
|E0 | = |E0 ∗ | =
r r r ∑ ∑ ⏐{ ⏐{ }⏐ ∑ }⏐ ∑ ⏐ E ∈ E0 ∗ : E ∩ Xn ∈ A(i) ⏐ ≤ ⏐ E ∈ Ln (r) : E ∩ Xn ∈ A(i) ⏐ = |A(i) |ai = a| A | i=0
i=0
i=0
A∈A
P. Borg / Discrete Mathematics 341 (2018) 1331–1335
and |E1 | = |B| =
∑
B∈B br −1
|E | = |E0 | + |E1 | ≤
= ∑
∑
B∈B b|B| .
a| A | +
A∈A
∑ B∈B
Clearly, |F0 | =
b|B| ≤
∑ A∈Xn
∑
a| A | +
A∈Xn a|A|
∑
and |F1 | =
∑
B∈Xn b|B| .
1335
We have
b|B| = |F0 | + |F1 | = |F |,
B∈Xn
and hence the result is proved. □ Remark 2.3. Note that if we assume Theorem 1.4, then we only need Case 2 of the proof of Theorem 1.5. Also note that the case r = 2n + 1 is proved in both Case 1 and Case 2. Thus, Case 1 allows us to restrict Case 2 to r > 2n + 1, in which case we do not have i = r − 1 when considering i ≤ 2n (as 2n < r − 1). Acknowledgments The author wishes to thank the anonymous referees for helpful remarks that led to an improvement in the presentation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
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