A generalization of the Bollobás set pairs inequality

The Bollobás set pairs inequality is a fundamental result in extremal set theory with many applications. In this paper, for n > k > t > 2, we consider a collection of k families Ai : 1 6 i 6 k where Ai = {Ai,j ⊂ [n] : j ∈ [n]} so that A1,i1∩· · ·∩Ak,ik 6= ∅ if and only if there are at least t distinct indices i1, i2, . . . , ik. Via a natural connection to a hypergraph covering problem, we give bounds on the maximum size βk,t(n) of the families with ground set [n]. Mathematics Subject Classifications: 05D05, 05D40, 05C65


Introduction
A central topic of study in extremal set theory is the maximum size of a family of subsets of an n-element set subject to restrictions on their intersections. Classical theorems in the area are discussed in Bollobás [2]. In this paper, we generalize one such theorem, known as the Bollobás set pairs inequality or two families theorem [3]: For convenience, we refer to a pair of families A and B satisfying the conditions of Theorem 1 as a Bollobás set pair. The inequality above is tight, as we may take the pairs (A i , B i ) to be distinct partitions of a set of size a + b with |A i | = a and |B i | = b for 1 i a+b a .
The latter inequality was proved for a = 2 by Erdős, Hajnal and Moon [5], and in general has a number of different proofs [11,12,14,17,18]. A geometric version was proved by Lovász [17,18], who showed that if A 1 , A 2 , . . . , A m and B 1 , B 2 , . . . , B m are respectively a-dimensional and b-dimensional subspaces of a linear space and dim(A i ∩ B j ) = 0 if and only if i, j ∈ [m] are distinct, then m a+b a .

Main Theorem
Theorem 1 has been generalized in a number of different directions in the literature [6,9,13,16,21,24]. In this paper, we give a generalization of Theorem 1 from the case of two families to k 3 families of sets with conditions on the k-wise intersections.  (1) and, for 2 j t, we define Using this notation, we generalize (1) as follows: be a surjection, and let We show in Section 2.1 that this inequality is tight for all k t = 2, but do not have an example to show that this inequality is tight for any t > 2.
For n k t 2, let β k,t (n) denote the maximum m such that there exists a Bollobás (k, t)-tuple of size m consisting of subsets of [n]. Then (1) gives β 2,2 (n) n n/2 which is tight for all n 2. Letting H(q) = −q log 2 q − (1 − q) log 2 (1 − q) denote the standard binary entropy function, we prove the following theorem: Theorem 3. For k 3 and large enough n, For k t 3 and large enough n, This determines log 2 β k,2 (n) up to a factor of order log 2 k and log 2 β k,t (n) up to a factor of order t 3 . We leave it as an open problem to determine the asymptotic value of (log 2 β k,t (n))/n as n → ∞ for any k 3 and t 2. A natural source for lower bounds on β k,t (n) comes from the probabilistic method -see the random constructions in Section 3.1 which establish the lower bounds in Theorem 3. To prove Theorem 3, we use a natural connection to hypergraph covering problems.

Covering hypergraphs
Theorem 1 has a wide variety of applications, from saturation problems [3,19] to covering problems for graphs [11,20], complexity of 0-1 matrices [23], geometric problems [1], counting cross-intersecting families [7], crosscuts and transversals of hypergraphs [24,25,26], hypergraph entropy [15,22], and perfect hashing [8,10]. In this section, we give an application of our main results to hypergraph covering problems. For a k-uniform hypergraph H, let f (H) denote the minimum number of complete k-partite k-uniform hypergraphs whose union is H. In the case of graph covering, a simple connection to the Bollobás set pairs inequality (1) may be described as follows. Let K n,n \ M denote the complement of a perfect matching M = {x i y i : 1 i n} in the complete bipartite graph K n,n with parts X = {x 1 , x 2 , . . . , x n } and Y = {y 1 , y 2 , . . . , y n }. If H 1 , H 2 , . . . , H m are complete bipartite graphs in a minimum covering of K n,n \ M , then let A i = {j : , it is straightforward to check that (A, B) is a Bollobás set pair, and Theorem 1 applies to give f (K n,n \M ) = min{m : m m/2 n}.
In a similar way, Theorem 2 applies to covering complete k-partite k-uniform hypergraphs. Let K n,n,...,n denote the complete k-partite k-uniform hypergraph with parts X i = {x ij : j ∈ [n]} for i ∈ [k]. Let H k,t (n) denote the subhypergraph consisting of hyperedges {x 1,i 1 , x 2,i 2 , . . . , x k,i k } such that at least t of the indices i 1 , i 2 , . . . , i k are distinct, and set f k,t (n) = f (H k,t (n)). Then there is a one-to-one correspondence between Bollobás (k, t)tuples of subsets of [m] and coverings of H k,t (n) with m complete k-partite k-graphs. We let β k,t (m) be the maximum size of a Bollobás (k, t)-tuple of subsets of [m], so that f k,t (n) = min{m : β k,t (m) n}.
This correspondence together with Theorem 2 will be exploited to prove which is partly an analog of (5). More generally, we prove the following theorem: Theorem 4. For k 3 and large enough n, the electronic journal of combinatorics 28(3) (2021), #P3.8 For k t 3 and large enough n, The bounds on β k,t (n) in Theorem (3) follow immediately from this theorem and (6). Equation (9) gives the order of magnitude for each t 3 as k → ∞, but for t = 2, Equation (8) has a gap of order log 2 k. From (7), we obtain β k,2 (n) n n/k . It is perhaps unsurprising that the asymptotic value of f k,t (n)/ log 2 n as n → ∞ is not known for any k > 2, since a limiting value of f (K k n )/ log 2 n is not known for any k > 2 -see Körner and Marston [15] and Guruswami and Riazanov [10].

Organization and notation
. For positive integers k n, let (n) (k) = (n)(n − 1) · · · (n − k + 1) denote the falling factorial. This paper is organized as follows. In Section 2, we prove Theorem 2. In Section 2.1, we construct a Bollobás (k, 2)-tuple which achieves equality in Theorem 2 and in Section 2.2, we construct a Bollobás (k, 2)-tuple which gives the lower bound in Equation (3). The upper bound on f k,t (n) in Theorem 4 comes from a probabilistic construction in Section 3.1, and the proof of the lower bound on f k,t (n) is given in Section 3.3; we prove (7) in Section 3.2.

Proof of Theorem 2
It follows that (A 1 (φ), . . . , A t (φ)) is a Bollobás set (t, t)-tuple and hence it suffices to prove Theorem 2 in the case where t = k. In this setting, surjections φ : [k] → [k] simply permute the k families and as such we suppress the notation of φ for the remainder of this section. One of the proofs of Theorem 1, given a Bollobás set pair, defines a collection of chains C i for i ∈ [m] and shows that these chains are necessarily disjoint. Similarly, given a Bollobás set (k, k)-tuple, we will define a collection of chains C σ for every ordered collection σ of (k − 1) distinct indices of [m] and show these chains are pairwise disjoint.
Proof. Seeking a contradiction, suppose there exists π ∈ C σ 1 ∩ C σ 2 . After relabeling, it suffices to consider the following five cases.
A similar argument yields the analog of Lemma 5 to the case where k 4.
. Seeking a contradiction, suppose there exists a π ∈ C σ 1 ∩ C σ 2 . Without loss of generality, which has no fixed points. As in Lemma 5, we want to show that there exists a w ∈ A h,σ 1 ∩ A k,σ 2 and consider two separate cases.
First, suppose that Next, suppose that σ 1 (h) = σ 2 (x) for some x. We now claim that x = 1. If h = 1, then this is trivial. If h > 1, then
Using Equation (10), Lemma 5, and Lemma 6, we are now able to prove Theorem 2 in the case where t = k. There are n! total permutations, and Lemma 5 and Lemma 6 yield that each of which appears in at most one of the sets C σ for σ ∈ [m] (k−1) . Hence, using |C σ | in Equation (10), and thus the result follows by dividing through by n!.

Sharpness of Theorem 2
We give a simple construction establishing the sharpness of Theorem 2 for k t = 2. Let n 4k and using addition modulo n, define for all j ∈ [k], we will show (A 1 , . . . , A k ) is a Bollobás (k, 2)-tuple. Since |A 1,i | = n − 1 and |A 2,i ∩ · · · ∩ A k,i | = 1, Theorem 2 with t = 2 and surjection φ : [k] → [2] with φ(1) = 1 and φ(i) = 2 for i = 1 gives By construction, for all i ∈ [n], A 1,i ∩ A 2,i ∩ · · · ∩ A k,i = ∅. It thus suffices to show these are the only empty k-wise intersections. To this end, for i = (i 1 , . . . , i k−1 ), define Proof. We proceed by induction on k where the result is trivial when k = 2. In the case Next, there is a y such that −(k − 2) y (k − 2) with i k−1 + (k − 2) = i k + y, and since n 4k, x + 2k − 4 = y with equality over Z and moreover i k−1 + (k − 2) = i k + (k − 2) over Z and hence i k = i k−1 . Removing these elements from each set, the result then follows by induction.

An Explicit Construction
Let k 3. An explicit construction of a Bollobás (k, 2)-tuple (A 1 , A 2 , . . . , A k ) where |A i | = 2 n and each A i consists of subsets of X for |X| = kn may be described as follows. Let I j := {x j,1 , x j,2 , . . . , x j,k } and consider X = I 1 · · · I n . Now, for each f : 3 Proof of Theorem 4

Upper bound on f k,t (n)
We wish to find a covering of H k,t (n) with complete k-partite k-graphs and assume the parts of H k,t (n) are X 1 , X 2 , . . . , X k . For each subset T of [k] of size t, consider the uniformly random coloring χ T : [n] → T . Given such a χ T , let Y i ⊂ X i be the vertices of color i for i ∈ T ; that is Y i := {x ij : χ(j) = i} and Y i = X i for i / ∈ T . Denote by H(T, χ) the (random) complete k-partite hypergraph with parts Y 1 , Y 2 , . . . , Y k , and note that H(T, χ) ⊂ H k,t (n). We place each H(T, χ) a total of N times independently and randomly where N = (t + 1)t t log 2 n (k − t + 1) log 2 e and produce k t N random subgraphs H(T, χ). For a set partition π of [k], let |π| denote the number of parts in the partition and index the parts by [|π|]. Given a set partition π = (P 1 , P 2 , . . . , P s ), let If U is the number of edges of H k,t (n) not in any of these subgraphs, then For sufficiently large n, we claim that E(U ) < 1, which implies there exists a covering of H k,t (n) with at most k t N complete k-partite k-graphs, as required. The following technical lemma states that f is a decreasing function in the set partition lattice, and that f (π, t) increases when we merge all but one element of a smaller part of π with a larger part of π: Lemma 8. Let k s t 2, and let π = (P 1 , P 2 , . . . , P s ) be a partition of [k].
The proof of Lemma 8 part (i) is in Appendix A and the proof of (ii) is similar to the proof of (i). By Lemma 8, a set partition of [k] into s parts which minimizes f (π, t) consists of one part of size k − s + 1 and s − 1 singleton parts and hence min{f (π, t) : In what follows, we denote a set partition of [k] into s parts which minimizes f (π, t) by π s .
For n large enough, and all s where t s k, we will show Replacing the numerator with its largest term and each term in denominator with its largest term, where S(k, s) is the Stirling number of the second kind. Taking n S(k, s), we will show in Appendix B that 1 Therefore, the index s = t maximizes the right hand side of Equation (13), and hence for our choice of N provided n kS(k, t). Thus,

Lower bound on f k,2 (n)
In this section, we show f k,2 (n) min{m : m m/k n}.
We use this inequality to give a lower bound on f k,2 (n) = m. First we observe Let ∂H denote the set of (k − 1)-tuples of vertices contained in some edge of a hypergraph H. Then Putting the above identities together, We note |∂H r ∩ ∂M | |V (H r )|/(k − 1), and therefore m r=1 It follows that Subject to the linear inequalities (18) and (22), the left side of (17) is minimized when (17) implies m m/k n, which gives (16).

Lower bound on f k,k (n)
Let H = {H 1 , H 2 , . . . , H m } be a minimal covering of H k,k (n) with complete k-partite kgraphs, so m = f (H k,k (n)). Given a k-partite k-graph H, consider its 2-shadow ∂ 2 (H) = {R ⊂ V (H) : |R| = k − 2, R ⊂ e for some e ∈ H}. Let ∂ 2 (H) = m i=1 ∂ 2 (H i ). Given R ∈ ∂ 2 (H) and H i ∈ H, let H i (R) := {e ∈ V (H i ) 2 : e ∪ R ∈ H i } be the possibly empty link graph of the edge R in the hypergraph H i and let V (H i (R)) be the set of vertices in the link graph. Observe that double counting yields An optimization argument yields |∂ 2 (H i )| is maximized when the parts of H i are of equal or nearly equal maximal size. Since |V (H i (R))| 2(n − k + 2), the right hand side of Equation (23) is bounded above by the electronic journal of combinatorics 28(3) (2021), #P3.8 generalization to Bollobás (k, t)-tuples for k 3 is equally interesting but wide open, as are potential generalizations to vector spaces -see Lovász [17,18].
• Orlin [20] proved that the clique cover number cc(K n \M ) of a complete graph K n minus a perfect matching M is precisely min{m : 2 m−1 m/2 n}. Theorem 4 yields lower bounds on the clique cover number of the complement of a perfect matching M in the complete k-uniform hypergraph K k n : Corollary 9. Let K k n \ M be the complement of a perfect matching in K k n . Then k log 2 n k log 2 (ke) .