aa r X i v : . [ m a t h . C O ] A p r Cross-Sperner families
D´aniel Gerbner a , ∗ Nathan Lemons b , ∗ Cory Palmer a , ∗ Bal´azs Patk´os a , †‡ Vajk Sz´ecsi b a Hungarian Academy of Sciences, Alfr´ed R´enyi Instituteof Mathematics, P.O.B. 127, Budapest H-1364, Hungary b Central European University, Department of Mathematicsand its Applications, N´ador u. 9, Budapest H-1051, Hungary
June 12, 2018
Abstract
A pair of families ( F , G ) is said to be cross-Sperner if there exists no pairof sets F ∈ F , G ∈ G with F ⊆ G or G ⊆ F . There are two ways to measurethe size of the pair ( F , G ): with the sum |F | + |G| or with the product |F | · |G| .We show that if F , G ⊆ [ n ] , then |F ||G| ≤ n − and |F | + |G| is maximal if F or G consists of exactly one set of size ⌈ n/ ⌉ provided the size of the groundset n is large enough and both F and G are non-empty. AMS Subject Classification : 05D05 keywords : extremal set systems, Sperner property
We use standard notation: [ n ] denotes the set of the first n positive integers, 2 S denotes the power set of the set S and (cid:0) Sk (cid:1) denotes the set of all k -element subsetsof S . The complement of a set F is denoted by F and for a family F we write F = { F : F ∈ F } .One of the first theorems in the area of extremal set families is that of Sperner[15], stating that if we consider a family F ⊆ [ n ] such that no set F ∈ F can contain ∗ Research supported by Hungarian National Scientific Fund, grant number: OTKA NK-78439 † Research supported by Hungarian National Scientific Fund, grant number: OTKA K-69062 andPD-83586 ‡ corresponding author, e-mail: [email protected] F ′ ∈ F , then the number of sets in F is at most (cid:0) n ⌊ n/ ⌋ (cid:1) and equality holds ifand only if F = (cid:0) [ n ] ⌊ n/ ⌋ (cid:1) or F = (cid:0) [ n ] ⌈ n/ ⌉ (cid:1) . Families satisfying the assumption of Sperner’stheorem are called Sperner families or antichains . The celebrated theorem of Erd˝os,Ko and Rado [6] asserts that if for a family G ⊆ (cid:0) [ n ] k (cid:1) we have G ∩ G ′ = ∅ for all G, G ′ ∈ G (families with this property are called intersecting ), then the size of G isat most (cid:0) n − k − (cid:1) provided 2 k ≤ n .There have been many generalizations and extensions both to the theorem ofSperner and to the result by Erd˝os, Ko and Rado (two excellent but not really recentsurveys are [4] and [5]). One such generalization is the following: a pair ( F , G ) offamilies is said to be cross-intersecting if for any F ∈ F , G ∈ G we have F ∩ G = ∅ . Cross-intersecting pairs of families have been investigated for quite a while andattracted the attention of many researchers [2, 3, 7, 8, 9, 10, 11, 12]. The presentpaper deals with the analogous generalization of Sperner families that has not beenconsidered in the literature. A pair ( F , G ) of families is said to be cross-Sperner ifthere exists no pair of sets F ∈ F , G ∈ G with F ⊆ G or G ⊆ F . There are twoways to measure the size of the pair ( F , G ): either with the sum |F | + |G| or with theproduct |F | · |G| . We will address both problems.Clearly, |F | + |G| ≤ n as by definition F ∩ G = ∅ . The sum 2 n can be obtained byputting F = ∅ , G = 2 [ n ] . Thus, when considering the problem of maximizing |F | + |G| we will assume that both F and G are non-empty.We can reformulate our problem in a rather interesting way. Let Γ n = ( V n , E n )be the graph with vertex set V n = 2 [ n ] and edge set E n = { ( F, G ) :
F, G ∈ V n F ( G or G ( F } . Then max {|F | + |G|} = 2 n − c (Γ n ), where c (Γ n ) denotes the vertexconnectivity of Γ n . Moreover, if we let F ( n, m ) = max {|G| : G ⊆ [ n ] , ∃F ⊆ [ n ] with |F | = m, ( F , G ) is cross-Sperner } , then, denoting by N Γ n ( U ) the neighborhood of U in Γ n , we have F ( n, m ) = 2 n − m − min {| N Γ n ( F ) | : F ⊆ V n , |F | = m } . Thus determining F ( n, m ) is equivalent to the isoperimetric problem for the graphΓ n . Let us mention that the cross-Sperner property of the pair ( F , G ) is equivalentto ( F , G ) being cross-intersecting and cross-co-intersecting, i.e. for any F ∈ F and G ∈ G we have F ∩ G = ∅ and F ∪ G = [ n ].The rest of the paper is organized as follows. In Section 2, we consider the problemof maximizing |F | + |G| and prove the following theorem. Theorem 1.1.
There exists an integer n such that if n ≥ n and the pair ( F , G ) iscross-Sperner with ∅ 6 = F , G ⊆ [ n ] , then |F | + |G| ≤ F ( n,
1) + 1 = 2 n − ⌈ n/ ⌉ − ⌊ n/ ⌋ + 2 , nd equality holds if and only if F or G consists of exactly one set S of size ⌊ n/ ⌋ or ⌈ n/ ⌉ and the other family consists of all subsets of [ n ] not contained in S and notcontaining S . In Section 3, we address the problem of maximizing |F | · |G| . Our result is thefollowing theorem.
Theorem 1.2. If n ≥ and ( F , G ) is cross-Sperner with F , G ⊆ [ n ] , then thefollowing inequality holds: |F ||G| ≤ n − . This bound is best possible as shown by F = { F ∈ [ n ] : 1 ∈ F, n / ∈ F } , G = { G ∈ [ n ] : n ∈ G, / ∈ G } . Finally, Section 4 contains some concluding remarks and open problems.
Before we start the proof of Theorem 1.1, let us introduce some notation and statea theorem that we will use in our proof. For a k -uniform family F ⊆ (cid:0) [ n ] k (cid:1) let ∆ F = { G ∈ (cid:0) [ n ] k − (cid:1) : ∃ F ∈ F , G ⊂ F } be the shadow of F . The following version of theshadow theorem is due to Lov´asz [13]. Theorem 2.1. [Lov´asz [13]]
Let
F ⊆ (cid:0) [ n ] k (cid:1) and let us define the real number x by |F | = (cid:0) xk (cid:1) . Then we have ∆ F ≥ (cid:0) xk − (cid:1) . For any F ∈ [ n ] we have N Γ n ( F ) = 2 | F | + 2 n −| F | − | F | = ⌈ n/ ⌉ . This proves F ( n,
1) = 2 n − ⌈ n/ ⌉ − ⌊ n/ ⌋ + 1 as stated in Theorem 1.1. Proposition 2.2.
If a pair ( F , G ) maximizes |F | + |G| , then both F and G are convexfamilies i.e. F ⊂ F ⊂ F , F , F ∈ F implies F ∈ F .Proof. If F, F , F are as above, then F can be added to F since any set containing F contains F and any subset of F is a subset of F .Let ( F , G ) be a pair of cross-Sperner families and let F and G be sets of minimumsize in F and G . Proposition 2.3. If | F | + | G | < ⌈ n/ ⌉ − , then |F | + |G| < F ( n, .Proof. No set containing F ∪ G can be a member of F or G .3s ( F , G ) is cross-Sperner if and only if ( F , G ) is cross-Sperner, by taking comple-ments (if necessary) and Proposition 2.3 we may and will assume that m := | F | ≥⌊ n/ ⌋ . Let us write F ∗ = { F ∈ F : F ( F } . Subsets of F are not in F by theminimality of F and by the cross-Sperner property they cannot be in G either, thusto prove Theorem 1.1 we need to show that there exist more than |F ∗ | many sets thatare not contained in F ∪ G and are not subsets of F . For any F ∗ ∈ F ∗ let us define B ( F ∗ ) = { F ∗ \ F ′ : F ′ ⊆ F , | F ∗ \ F ′ | < m } . Clearly, for any F ∗ , F ∗ ∈ F ∗ we have B ( F ∗ ) ∩ B ( F ∗ ) = ∅ as they already differ outside F . By definition, no set in B := ∪ F ∗ ∈F ∗ B ( F ∗ ) is a subset of F . We have B ∩ F = ∅ as all sets in B have size smaller than m and B ∩ G = ∅ by the cross-Sperner property.Thus to prove Theorem 1.1 it is enough to show that |F ∗ | < |B| .Note the following three things: • | B ( F ∗ ) | = P mi = | F ∗ \ F | +1 (cid:0) mi (cid:1) , • F ∗∗ = { F ∗ \ F : F ∗ ∈ F ∗ } is downward closed as F and F ∗ are convex, • |F ∗∗ | = |F ∗ | .Therefore the following lemma finishes the proof of Theorem 1.1 by choosing A = F ∗∗ , k = m and n ′ = n − | F | . Lemma 2.4.
Let ∅ 6 = A ⊆ [ n ′ ] be a downward closed family and k ≥ n ′ / . Then if n ′ is large enough, the following holds |A| < X A ∈A k X i = | A | +1 (cid:18) ki (cid:19) . (1) Proof.
Let a i = |{ A ∈ A : | A | = i }| and w ( j ) = P ki = j +1 (cid:0) ki (cid:1) . Then we can formulate(1) in the following way: n ′ X j =0 a j < n ′ X j =0 a j w ( j ) . (2)Let x be defined by a k − = (cid:0) xk − (cid:1) . By Theorem 2.1 if j < k − a j ≥ (cid:0) xj (cid:1) . Ifwe replace a j by (cid:0) xj (cid:1) in (2), then the LHS decreases by a j − (cid:0) xj (cid:1) and the RHS decreasesby ( a j − (cid:0) xj (cid:1) ) w ( j ), which is larger. If j > k −
1, then a j ≤ (cid:0) xj (cid:1) again by Theorem 2.1.If we replace a j by (cid:0) xj (cid:1) in (2), then the LHS increases while the RHS does not change(as for j ≥ k we have w ( j ) = 0). Hence it is enough to prove4 ′ X j =0 (cid:18) xj (cid:19) < n ′ X j =0 (cid:18) xj (cid:19) w ( j ) . (3)First we prove (3) for x = n ′ . In this case the LHS is 2 n ′ while the RHS ismonotone increasing in k , thus it is enough to prove for k = ⌈ n/ ⌉ . We will estimatethe RHS from below by considering only one term of the sum. Clearly, (cid:0) n ′ j (cid:1) w ( j ) ≥ (cid:0) n ′ j (cid:1)(cid:0) kj +1 (cid:1) ≥ (cid:0) n ′ j (cid:1)(cid:0) n ′ / j +1 (cid:1) . Let us write j = αn ′ for some 0 ≤ α ≤ /
3. Then by Stirling’sformula we obtain (cid:18) n ′ j (cid:19)(cid:18) n ′ / j + 1 (cid:19) = (cid:18) n ′ αn ′ (cid:19)(cid:18) n ′ / αn ′ + 1 (cid:19) = Θ n ′ (cid:18) α α (1 − α ) − α / (1 / − α ) / − α (cid:19) n ′ ! . The value of the fraction in parenthesis is larger than 2 for, say, α = 2 /
9, thus (3)holds if n ′ is large enough and x = n ′ .To prove (3) for arbitrary x , let c = (cid:0) xk − (cid:1) / (cid:0) n ′ k − (cid:1) . If j > k −
1, then c > (cid:0) xj (cid:1) / (cid:0) n ′ j (cid:1) ,while if j < k −
1, then c < (cid:0) xj (cid:1) / (cid:0) n ′ j (cid:1) . By the x = n ′ case we know n ′ X j =0 c (cid:18) n ′ j (cid:19) < n ′ X j =0 c (cid:18) n ′ j (cid:19) w ( j ) . (4)Let us replace c (cid:0) n ′ j (cid:1) by (cid:0) xj (cid:1) in this inequality. If j > k −
1, then the LHS decreasesand the RHS does not change. If j = k − c . If j < k −
1, both sides increase, and the RHS increases more as w ( j ) ≥ ≤ j ≤ k −
1. Hence the inequality holds and gives back (3), which finishes theproof of the lemma.We believe that Theorem 1.1 is valid for all n , but unfortunately Lemma 2.4 failsfor small values of n . In this section we prove Theorem 1.2. Our main tool will be the following specialcase of the Four Functions Theorem of Ahlswede and Daykin [1]. To state theirresult for any pair A , B of families let us write A ∧ B = { A ∩ B : A ∈ A , B ∈ B} and A ∨ B = { A ∪ B : A ∈ A , B ∈ B} . Theorem 3.1. [Ahlswede-Daykin, [1]]
For any pair A , B of families we have |A||B| ≤ |A ∧ B||A ∨ B| .
5o prove Theorem 1.2 we will need the following lemma.
Lemma 3.2. If ( F , G ) is a pair of cross-Sperner families, then the families F , G , F ∧ G and
F ∨ G are pairwise disjoint.Proof. F and G are disjoint as some set F ∈ F ∩ G is a subset of itself and thuscontradicts the cross-Sperner property. F and G are both disjoint from F ∧ G and
F ∨ G as F ∩ G ⊆ F, G and
F, G ⊆ F ∪ G . Finally, F ∧ G and
F ∨ G are disjoint as F ∩ G = F ∪ G would imply F ⊆ G .Now we are able to prove Theorem 1.2. Proof.
Let ( F , G ) be a cross-Sperner pair of families. Clearly, if |F | + |G| ≤ n − ,then the statement of the theorem holds. But if |F | + |G| > n − , then by Lemma 3.2we have |F ∧ G| + |F ∨ G| < n − and thus by Theorem 3.1 we obtain |F ||G| ≤|F ∧ G||F ∨ G| ≤ n − . Corollary 3.3.
For n ≥ , we have F ( n, n − ) = 2 n − . One might wonder whether it changes the situation if we allow sets to belong to both F and G and we modify the definition of cross-Sperner families so that only pairs F ∈ F , G ∈ G with F ( G or G ( F are forbidden. It is easy to see that thesituation is the same when considering |F | + |G| . To prove that |F | + |G| ≤ n letus write C = F ∩ G and if it is not empty, then D ( C ) := { C \ C ′ : C, C ′ ∈ C} isdisjoint both from F and G and a result by Marica and Sch¨onheim [14] tells us that | D ( C ) | ≥ |C| . Note that the proof of Theorem 1.1 works in this case as well givingthe upper bound |F | + |G| ≤ F ( n,
1) + 2.Although F ( n, m ) is not known for most values, it is natural to generalize theproblem to k -tuples of families: F , F , ..., F k is said to be cross-Sperner if for any1 ≤ i < j ≤ k there is no pair F ∈ F i and F ′ ∈ F j with F ⊆ F ′ or F ′ ⊆ F . One canconsider the problems of maximizing P ki =1 |F i | and Q ki =1 |F i | . In the former case weneed the extra assumption that all F i are non-empty as otherwise the trivial upperbound 2 n is tight.When maximizing the sum, it is natural to conjecture that in the best possibleconstruction all but one family consists of one single set. By the cross-Sperner prop-erty, these sets together must form a Sperner family, therefore it might turn out tobe useful to introduce F ∗ ( n, m ) = max {|G| : G ⊆ [ n ] , ∃F ⊆ [ n ] with |F | = m, ( F , G ) is cross-Sperner, F is Sperner } . roblem 4.1. Under what conditions is it true that if F , F , ..., F k form a k -tupleof non-empty cross-Sperner families, then k X i =1 |F i | ≤ k − F ∗ ( n, k − |F i | , by Theorem 1.2 one obtains that k Y i =1 |F i | = Y ≤ i R. Ahlswede, D. Daykin , An inequality for the weights of two families ofsets, their unions and intersections, Probability Theory and Related Fields, (1978), 183–185.[2] C. Bey On cross-intersecting families of sets, Graphs and Combinatorics, (2005), 161–168.[3] B. Bollob´as , On generalized graphs, Acta Mathematica Academiae Scientar-ium Hungaricae (1965) (34), 447–452.[4] M. Deza, P. Frankl , Erd˝os–Ko–Rado theorem - 22 years later, SIAM J.Algebraic Discrete Methods, (1983), 419–431.75] K. Engel , Sperner Theory , Encyclopedia of Mathematics and its Applications,65. Cambridge University Press, Cambridge, 1997. x+417 pp.[6] P. Erd˝os, C. Ko, R. Rado , Intersection theorems for systems of finite sets,Quart. J. Math. Oxford, (1961), 313–318.[7] P. Frankl, N. Tokushige , Some best possible inequalities concerning cross-intersecting families, J. of Comb. Theory, Ser. A, (1992) 87-97.[8] P. Frankl, N. Tokushige , Some inequalities concerning cross-intersectingfamilies, Combinatorics, Probability and Computing, (1998), 247–260.[9] Z. F¨uredi , Cross-intersecting families of finite sets, J. of Comb. Theory, Ser. A, (1995), 332–339.[10] ´A. Kisv¨olcsey , Exact bounds on cross-intersecting families, Graphs and Com-binatorics, (2001), 275–287.[11] ´A. Kisv¨olcsey , Weighted cross-intersecting families, Discrete Mathematics, (2008), 2247–2260.[12] P. Keevash, B. Sudakov , On a restricted cross-intersection problem, J. ofComb. Theory, Ser. A, (2006), 1536–1542.[13] L. Lov´asz , Combinatorial problems and exercises , second edition, North-Holland Publishing Co., Amsterdam, 1993, 635 pp.[14] J. Marica, J. Sch¨onheim , Differences of sets and a problem of Graham, Can.Math. Bull. (1969), 635–637.[15] E. Sperner , Ein Satz ¨uber Untermenge einer endlichen Menge, Math Z.,27