Note: Union-closed families with small average overlap densities
aa r X i v : . [ m a t h . C O ] D ec Note: Union-closed families with small average overlap densities.
David Ellis ∗ Abstract
In this very short note, we point out that the average overlap density of a union-closed family F of subsets of { , , . . . , n } may be as small as Θ((log log |F| ) / (log |F| )), for infinitely manypositive integers n . If X is a set, a family F of subsets of X is said to be union-closed if the union of any two sets in F is also in F . The celebrated Union-Closed Conjecture (a conjecture of Frankl [2]) states that if X is a finite set and F is a union-closed family of subsets of X (with F 6 = {∅} ), then there exists anelement x ∈ X such that x is contained in at least half of the sets in F . Despite the efforts of manyresearchers over the last forty-five years, and a recent Polymath project [5] aimed at resolving it,this conjecture remains wide open. It has only been proved under very strong constraints on theground-set X or the family F ; for example, Balla, Bollob´as and Eccles [1] proved it in the casewhere |F | ≥ | X | ; more recently, Karpas [3] proved it in the case where |F | ≥ ( − c )2 | X | for asmall absolute constant c >
0; and it is also known to hold whenever | X | ≤
12 or |F | ≤
50, fromwork of Vuˇckovi´c and ˇZivkovi´c [8] and of Roberts and Simpson [7].In 2016, a Polymath project [5] was convened to tackle the Union-Closed Conjecture. Whileit did not result in a proof of the conjecture, several interesting related conjectures were posed.Among them was the ‘average overlap density conjecture’.If X is a finite set and F ⊂ P ( X ) with F 6 = ∅ , we define the abundance of x (with respect to F ) by γ x = |{ A ∈ F : x ∈ A }| / |F | , i.e., γ x is the probability that a uniform random member of F contains x . A natural first quantity to consider, in trying to prove the Union-Closed Conjecture,is the average abundance of a uniform random element of the ground set, i.e., E x ∈ X [ γ x ]; if thisquantity were always at least 1 /
2, the Union-Closed Conjecture would immediately follow. Amoment’s thought shows that this is false, however, e.g. by considering the union-closed family {∅ , { } , { , , }} ⊂ P ( { , , } ), which has average abundance 4 /
9. Similarly, for any n ∈ N , theunion-closed family F = {∅ , { } , { , } , . . . , { , , . . . , ⌊√ n ⌋} , { , , , . . . , n }} ⊂ P ( { , , . . . , n } ) hasaverage abundance Θ(1 / √ n ) = Θ(1 / |F | ).It is natural to consider the expected abundance of a random element of the ground-set X chosen according to other (non-uniform) distributions on X . The following was considered in the ∗ School of Mathematics, University of Bristol. average overlap density
AOD( F ) of F to be the expected valueof γ x , where x is a uniform random element of a uniform random nonempty member of F :AOD( F ) := 1 |F \ {∅}| X A ∈F\{∅} | A | X x ∈ A γ x = 1 |F \ {∅}| X A ∈F\{∅} | A | X x ∈ A |{ B ∈ F : x ∈ B }||F | = 1 |F \ {∅}| X A ∈F\{∅} |F | X B ∈F | A ∩ B || A | ! = E A ∈F\{∅} E B ∈F (cid:20) | A ∩ B || A | (cid:21) . (1)(The first and second expectations in (1) are of course over a uniform random element of F \ {∅} ,and a uniform random element of F , respectively.) The last equality justifies the ‘average overlap’terminology. The average overlap density conjecture stated that if X is a finite set, and F is aunion-closed family of subsets of X with F 6 = ∅ and F 6 = {∅} , then the average overlap density of F is at least 1 /
2. Clearly, it would immediately imply the Union-Closed Conjecture.Unfortunately, the average overlap density conjecture was quickly shown to be false (duringthe Polymath project [6]); an infinite sequence of union-closed families F n ⊂ P ( { , , . . . , n } ) wasconstructed with AOD( F n ) = 7 /
15 + o (1) as n → ∞ . However, the following weakening of theaverage overlap density conjecture remained open. Conjecture 1.
There exists an absolute positive constant c > such that the following holds. Let n ∈ N and let F ⊂ P ( { , , . . . .n } ) be union-closed with F 6 = {∅} . Then the average overlap densityof F is at least c . Conjecture 1 would immediately imply the weakening of the Union-Closed Conjecture where 1 / c . In this note, we give an example of a union-closedfamily F of subsets of { , , . . . , n } whose average overlap density is Θ((log log |F | ) / (log |F | )) forinfinitely many positive integers n , disproving Conjecture 1 in a strong sense. It follows from anold result of Knill [4] that if F ⊂ P ( { , , . . . , n } ) is union-closed, then there exists x ∈ { , , . . . , n } with abundance γ x = Ω(1 / (log |F | )), so the average overlap density can, in the best-case scenario,only be used to improve this lower bound by a factor of Θ(log log |F | ). For n ∈ N , we write [ n ] := { , , . . . , n } for the standard n -element set, and if G ⊂ P ( X ), the union-closed family generated by G is defined to be the smallest union-closed family of subsets of X that contains G .Let k, m, s ∈ N with s ≤ k − m ≥
2, and let n = km . Partition [ n ] into m sets B , . . . , B m with | B i | = k for all i ; in what follows, we will refer to the B i as ‘blocks’. For each i ∈ [ m ], choosea subset T i ⊂ B i with | T i | = s , and let T = ∪ mi =1 T i . Now let F ⊂ P ([ n ]) be the union-closed familygenerated by { B i ∪ { j } : i ∈ [ m ] , j ∈ T } . Note that every set in F contains at least one block.The number of sets in F containing exactly one block is m ( m − s , and in general, for each j ∈ [ m ],2umber N j of sets in F containing exactly j blocks is (cid:0) mj (cid:1) ( m − j ) s , so N := |F | = m X j =1 N j = 2 ( m − s m X j =1 (cid:18) mj (cid:19) − ( j − s . For each j ∈ [ m ], define p j := N j /N ; this is of course the probability that a uniform randommember of F contains exactly j blocks. We note that p j +1 p j = N j +1 N j = m − jj + 1 2 − s ≤ m − s ∀ j ∈ [ m − . Write τ := m − s . For any x ∈ [ n ] \ T , we clearly have γ x = 1 m m X j =1 jp j , since the conditional probability that x is contained in a random member A of F , given that A contains exactly j blocks, is j/m . We have p j ≤ τ j − p for all j ∈ [ m ], and therefore for any x ∈ [ n ] \ T , we have1 m ≤ γ x ≤ m (1 + 2 τ + 3 τ + . . . + mτ m − ) ≤ m (1 + 4 τ ) ≤ m , provided τ = m − s ≤ /
4. Now, every member A of F contains at least one block, so for anymember A of F , the probability a uniform random element of A is in T , is at most msk . Crudely,we have 1 / ≤ γ x ≤ x ∈ T , since A A ∪ { x } is an injection from { A ∈ F : x / ∈ A } to { A ∈ F : x ∈ A } , for any x ∈ T . Hence, we have1 m ≤ AOD( F ) ≤ (cid:16) − msk (cid:17) · m + msk · ≤ m + m sn , (2)again provided τ = m − s ≤ /
4. Now we wish to minimize the right-hand side of (2), subject tothe constraint m − s ≤ /
4; clearly the optimal choice is to take s = ⌈ log m ⌉ + 2, which yields1 m ≤ AOD( F ) ≤ m + m log mn + O ( m /n ) . (3)It is clear that the optimal choice of m to minimize the right-hand side of (3) is Θ(( n/ (log n )) / ),yielding AOD( F ) = Θ(((log n ) /n ) / ). Since, with these choices, log |F | = Θ( n / (log n ) / ), wehave AOD( F ) = Θ((log log |F | ) / (log |F | )). Note also that the average abundance of a uniformrandom element of [ n ] (with respect to F ) satisfies E x ∈ [ n ] [ γ x ] = Θ((log log |F | ) / (log |F | )).We remark that the family F constructed above does not separate the points of [ n ]. (We saya family F ⊂ P ([ n ]) separates the points of [ n ] if for any i = j ∈ [ n ] there exists A ∈ F such that | A ∩ { i, j }| = 1. It is easy to see that, in attempting to prove the Union-Closed Conjecture, we mayassume that the union-closed family in question separates the points of the ground set, and thisassumption was adopted for much of the Polymath project [5].) However, it is easy to see that theunion-closed family F ∪ { [ n ] \ { j } : j ∈ [ n ] } has asymptotically the same average overlap densityas F (and asymptotically the same average abundance as F ), and does separate the points of [ n ].3 eferences [1] I. Balla, B. Bollob´as and T. Eccles, Union-closed families of sets. J. Combin. Theory (SeriesA) , 120 (2013), 531–544.[2] D. Duffus, in: I. Rival (Ed.),
Graphs and Order . Reidel, Dordrecht, Boston, 1985, p. 525.[3] I. Karpas, Two Results on Union-Closed Families. Preprint, August 2017. arXiv:1708.01434.[4] E. Knill, Graph generated union-closed families of sets. Manuscript, September 1994.arXiv:9409215.[5]
Polymath11: Frankl’s Union-Closed Conjecture.
1: Strengthenings, variants, potentialcounterexamples. https://gowers.wordpress.com/2016/01/29/func1-strengthenings-variants-potential-counterexamples/ .[6]
Polymath11: Frankl’s Union-Closed Conjecture.
2: More examples. https://gowers.wordpress.com/2016/02/08/func2-more-examples/ .[7] I. Roberts and J. Simpson, A note on the union-closed sets conjecture.
Australas. J. Combin. ,47 (2010), 265–267.[8] B. Vuˇckovi´c and M. ˇZivkovi´c, The 12-element case of Frankl’s conjecture.