Small Sets in Union-Closed Families

Our aim in this note is to show that, for any $\epsilon>0$, there exists a union-closed family $\mathcal F$ with (unique) smallest set $S$ such that no element of $S$ belongs to more than a fraction $\epsilon$ of the sets in $\mathcal F$. More precisely, we give an example of a union-closed family with smallest set of size $k$ such that no element of this set belongs to more than a fraction $(1+o(1))\frac{\log_2 k}{2k}$ of the sets in $\mathcal F$. We also give explicit examples of union-closed families containing `small' sets for which we have been unable to verify the Union-Closed Conjecture.


Introduction
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 Union-Closed Conjecture (a conjecture of Frankl [5]) states that if X is a finite set and F is a union-closed family of subsets of X (with F = {∅}), then there exists an element x ∈ X such that x is contained in at least half of the sets in F. Despite the efforts of many researchers over the last forty-five years, and a recent Polymath project [7] aimed at resolving it, this conjecture remains wide open. It has only been proved under very strong constraints on the ground-set X or the family F; for example, Balla, Bollobás and Eccles [3] proved it in the case where |F| ≥ 2 3 2 |X| ; more recently, Karpas [6] proved it in the case where |F| ≥ ( 1 2 − c)2 |X| for a small absolute constant c > 0; and it is also known to hold whenever |X| ≤ 12 or |F| ≤ 50, from work of Vučković and Živković [11] and of Roberts and Simpson [9]. Note that the Union-Closed Conjecture is not even known to hold in the weaker form where we replace the fraction 1/2 by any other fixed ǫ > 0. 1 For general background and a wealth of further information on the Union-Closed Conjecture see the survey of Bruhn and Schaudt [4].
As usual, if X is a set we write P(X) for its power-set. If X is a finite set and F ⊂ P(X) with F = ∅, we define the frequency of x (with respect to F) to be γ x = |{A ∈ F : x ∈ A}|/|F|, i.e., γ x is the proportion of members of X that contain x. If a union-closed family contains a 'small' set, what can we say about the frequencies in that set?
If a union-closed family F contains a singleton, then that element clearly has frequency at least 1/2, while if it contains a set S of size 2 then, as noted by Sarvate and Renaud [10], some element of S has frequency at least 1/2. However, they also gave an example of a union-closed family F whose smallest set S has size 3 and yet where each element of S has frequency below 1/2. Generalising a construction of Poonen [8], Bruhn and Schaudt [4] gave, for each k ≥ 3, an example of a union-closed family with (unique) smallest set of size k and with every element of that set having frequency below 1/2.
However, in these and all other known examples, there is always some element of a minimal-size set having frequency at least 1/3. So it is natural to ask if there is really a constant lower bound for these frequencies.
Our aim in this note is to show that this is not the case.
Theorem 1. For any positive integer k, there exists a union-closed family in which the (unique) smallest set has size k, but where each element of this set has frequency (All logarithms in this paper are to base 2. Also, as usual, the o(1) denotes a function of k that tends to zero as k tends to infinity.) The proof of Theorem 1 is by an explicit construction.
Theorem 1 is asymptotically sharp, in view of results of Wójcik [12] and Balla [2]: Wójcik showed that if S is a set of size k ≥ 1 in a finite union-closed family, then the average frequency of the elements in S is at least c k , where k · c k is defined to be the minimum average set-size over all union-closed families on the ground-set [k], and Balla showed that c k = (1 + o(1)) log k 2k , confirming a conjecture of Wójcik from [12].
Remarkably, there are union-closed families containing small sets, even sets of size 3, for which we have been unable to verify the Union-Closed Conjecture. We give some examples at the end of the paper.

Proof of main result
For our construction, we need the following 'design-theoretic' lemma.
Lemma 2. For any positive integers k > t there exist infinitely many positive integers d such that t divides dk and the following holds. If X is a set of size dk/t, then there exists a family A = {A 1 , . . . , A k } of k d-element subsets of X, such that each element of X is contained in exactly t sets in A, and for 2 ≤ r ≤ t, any r distinct sets in A have intersection of size Proof. Let q be a positive integer, and set d = k−1 t−1 q t ; we will take |X| = k t q t . Partition [qk] into k sets, B 1 , B 2 , . . . , B k say, each of size q; we call these sets 'blocks'. We let X be the set of all t-element subsets of [qk] that contain at most one element from each block. For each i ∈ [k] we let A i be the family of all sets in X that contain an element from the block B i . Clearly, |A i | = k−1 t−1 q t = d for each i ∈ [k], and each element of X appears in exactly t of the A i . Also, for example A i ∩ A j consists of all sets in X that contain both an element from the block B i and an element from the block B j , so It is easy to check that the other intersections also have the claimed sizes.
We remark that, in what follows, it is vital that the integer d in Lemma 2 can be taken to be arbitrarily large as a function of k and t.
Proof of Theorem 1. We define n = dk/t + k, we take d ∈ N as in the above lemma, and we let X = [dk/t]; the claim yields a family A = {A 1 , . . . , A k } of k d-element subsets of X = [dk/t] such that each element of [dk/t] is contained in exactly t of the sets in A, and for any 2 ≤ r ≤ t, any r distinct sets in A have intersection of size Write m = dk/t. We take F ⊂ P([n]) to be the smallest union-closed family containing the k-element set {m + 1, . . . , m + k} and all sets of the form For brevity, we write S 0 = {m + 1, m + 2, . . . , m + k}. We will show that each element of S 0 has frequency provided t and d are chosen to be appropriate functions of k; moreover, with these choices, S 0 will be the smallest set in F. Clearly, F contains S 0 , all sets of the form S 0 ∪(X\{x}) for x ∈ X, all sets of the form R∪X where R is a nonempty subset of S 0 , and finally all sets of the form R∪(X \{x}), where R = {m+i 1 , . . . , m+i r } is a nonempty r-element subset of S 0 and It is easy to see that the family F contains no other sets.
It follows that On the other hand, the number of sets in F that contain the element m + 1 is equal to It follows that the frequency of m + 1 (or, by symmetry, of any other element of S 0 ) equals To (asymptotically) minimise this expression, we take t = ⌊log k⌋ and d → ∞ (for fixed k); this yields a union-closed family in which the (unique) smallest set (namely S 0 ) has size k, and every element of that set has frequency proving the theorem.

An open problem
We now turn to some explicit examples of union-closed families containing small sets for which we have been unable to establish the Union-Closed Conjecture. For simplicity, we concentrate on the most striking case, when the family contains a set of size 3, and indeed is generated by sets of size 3.
Our families live on ground-set Z 2 n , the n × n torus. Perhaps the most interesting case is when n is prime. In that case we may assume that R = {0, 1, r} for some r, and so one feels that the verification of the Union-Closed Conjecture should be a triviality, but it seems not to be. Note that all the families in Question 3 are transitive families, in the sense that all points 'look the same', so that the Union-Closed Conjecture is equivalent to the assertion that the average size of the sets in the family is at least n 2 /2.
We mention that the corresponding result in Z n (in other words, the union-closed family on groundset Z n generated by translates of R) is known to hold: this is proved in [1].
We have verified the special case of Question 3 where R = {0, 1, 2}. A sketch of the proof is as follows. Assume that n ≥ 6, and let F ⊂ P(Z 2 n ) be the union-closed family generated by all translates of {0, 1, 2}×{0} and of {0}×{0, 1, 2} (we call these translates 3-tiles, for brevity). Let C = {0, 1, 2, 3} 2 , a 4×4 square. Consider the bipartite graph H = (X , Y) with vertex-classes X and Y, where X consists of all subsets of C with size less than 8 that are intersections with C of sets in F, Y consists of all subsets of C with size greater than 8 that are intersections with C of sets in F, and we join S ∈ X to S ′ ∈ Y if |S ′ | + |S| ≥ 16 and S ′ = S ∪ U for some union U of 3-tiles that are contained within C. It can be verified (by computer) that H has a matching m : X → Y of size |X | = 16520. Such a matching m gives rise to an injection with the property that |S ∩ C| + |f (S) ∩ C| ≥ |C| for all S ∈ F with |S ∩ C| < |C|/2. It follows that a uniformly random subset of F has intersection with |C| of expected size at least |C|/2, which in turn implies that there is an element of C with frequency at least 1/2 (and in fact, since F is transitive, every element has frequency at least 1/2).
We remark also that it would be nice to find a non-computer proof of the above result.