Coloring cross-intersecting families

Intersecting and cross-intersecting families usually appear in extremal combinatorics in the vein of the Erd{\H o}s--Ko--Rado theorem. On the other hand, P.~Erd{\H o}s and L.~Lov{\'a}sz in the noted paper~\cite{EL} posed problems on coloring intersecting families as a restriction of classical hypergraph coloring problems to a special class of hypergraphs. This note deals with the mentioned coloring problems stated for cross-intersecting families.


Introduction
Intersecting families in extremal combinatorics appeared in [4], and a large branch of extremal combinatorics starts from this paper. Then P. Erdős and L. Lovász in [6] introduced several problems on coloring intersecting families (cliques in the original notation), i. e. hypergraphs without a pair of disjoint edges. Obviously, an intersecting family could have chromatic number 2 or 3 only; the main interest refers to chromatic number 3. Unfortunately, there is no "random" example of such family, so the set of known intersecting families with chromatic number 3 is very poor.
Cross-intersecting families were introduced to study maximal and almost-maximal intersecting families (the notation appears in [13]). Definition 1.2. Cross-intersecting family is a hypergraph H = (V, E = A ∪ B) such that every a ∈ A intersects every b ∈ B, and A, B are not empty.
Also, the Hilton-Milner theorem [10] and the Frankl theorem [7] should be noted. Recently a general approach to mentioned problems was introduced by A. Kupavskii and D. Zakharov [12] (the reader can also see this paper for a survey).

The chromatic number
We are interested in vertex colorings of cross-intersecting families. Coloring is proper if there are no monochromatic edges. Chromatic number is the minimal number of colors that admits a proper coloring. First, note that a cross-intersecting family could have an arbitrarily large chromatic number. However, under a natural assumption (note that it holds for any n-uniform hypergraph) a chromatic number of a cross-intersecting family is bounded. Proof. Let us color a ∩ b in color 1, a \ b in color 2, b \ a in color 3 and all other vertices in color 4. One can see that the coloring is proper because both a and b have no subedge.
It turns out, that if there is no pair e 1 , e 2 ∈ E such that e 1 ⊂ e 2 and every edge has a size of at least 3, then the cross-intersecting family can have chromatic number 2 or 3 only. Moreover, the following theorem holds.

Maximal number of edges
It turns out that the maximal number of edges in a "nontrivial" n-uniform intersecting family is bounded. There are two ways to formalize the notion "nontrivial". The first one is to say that χ(H) ≥ 3 (denote the corresponding maximum by M (n)). The second one says that H is nontrivial if and only if τ (H) = n (denote the corresponding maximum by r(n)), where τ (H) is defined below. Definition 1.8. Let H = (V, E) be a hypergraph. The covering number (also known as transversal number or blocking number) of H is the smallest integer τ (H) that there is a set A ⊂ V such that every e ∈ E intersects A and |A| = τ .

Upper bounds.
Obviously, M (n) ≤ r(n). P. Erdős and L. Lovász proved in [6] that r(n) ≤ n n (one can find slightly better bound in [2]). The best current upper bound is r(n) ≤ cn n−1 (see [1]). Surprisingly, we can prove a very similar statement for cross-intersecting families. Let us introduce a "nontriviality" notion for crossintersecting families.

Definition 1.9. Let us call a cross-intersecting family
Note that if an n-uniform intersecting family H = (V, E) has τ (H) = n then (V, E, E) is a critical crossintersecting family. Then max(|A|, |B|) ≤ n n .

Lower bounds.
L. Lovász conjectured that M (n) = [(e − 1)n!] (an example was constructed in [6]). This was disproved by P. Frankl, K. Ota and N. Tokushige [8]. They have provided an explicit example of an n-uniform hypergraph H with τ (H) = n and

Examples
Unlike the case of intersecting families there is a method of constructing a large set of (critical) crossintersecting families with chromatic number 3, based on percolation. This method makes it possible to construct a cross-intersecting family from a random planar triangulation.    Let us take an (n − 1)-uniform simple hypergraph H 0 = (V 0 , E 0 ) such that χ(H) = 3 (see [6,11]  Suppose that |e| > 2. It means that there are different s, t 1 , t 2 ∈ e. Note that {r, t 1 } ∈ A since {q, s} ∈ A, so {q, t 2 } ∈ A. Thus every edge {x, t} ∈ A for every choice x ∈ {q, r} and t ∈ e. Obviously, we have listed all edges of the hypergraph, so we proved the claim in this case. Note also that the set of colors in {q, r} Then H is a flower with k petals with core W if τ (F W ) ≥ k.
The following Lemma was proved by J. Håstad, S. Jukna and P. Pudlák [9]. We provide its proof for the completeness of presentation. Proof. Induction on n. The basis n = 1 is trivial. Now suppose that the lemma is true for n− 1 and prove it for n. If τ (H) ≥ k then H itself is a flower with at least k petals (and an empty core). Otherwise, some set of size k − 1 intersects all the edges of H, and hence, at least |E|/(k − 1) of the edges must contain some vertex x. The hypergraph edges, each of cardinality at most n − 1. By the induction hypothesis, H {x} contains a flower with k petals and some core Y . Adding the element x back to the sets in this flower, we obtain a flower in H with the same number of petals and the core Y ∪ {x}. Now let us prove Theorem 1.10. Suppose the contrary, i. e. that, without loss of generality, |A| ≥ n n + 1. Then by Lemma the hypergraph (V, A) contains a flower with n + 1 petals. It means that every b ∈ B intersects the core of the flower, and H is not critical. A contradiction.

Open questions
The most famous problem in hypergraph coloring is to determine the minimal number of edges in an nuniform hypergraph with χ(H) = 3 (it is usually denoted by m(n)). The best known bounds ( [5,14,3]) are c n ln n 2 n ≤ m(n) ≤ e · ln 2 4 n 2 2 n (1 + o(1)).
P. Erdős and L. Lovász in [6] posed the same question for the class of intersecting families. Even though the intersecting condition is very strong, it does not provide a better lower bound. On the other hand, the upper bound in (2) is probabilistic, so it does not work for intersecting families. So the asymptotically best upper bound is 7 n−1 2 for n = 3 k , which is given by iterated Fano plane. Another question is to determine the minimal size a(n) of the largest intersection in an n-uniform intersecting family. The best bounds at this time are n log 2 n ≤ a(n) ≤ n − 2.
Studying the mentioned problems for cross-intersecting families is also of interest.
Recall that Example 1.15 shows that Theorem 1.10 is tight. On the other hand, max min(|A|, |B|) over all cross-intersecting families with chromatic number 3 is unknown. Obviously, one may take the example (V, E) by P. Frankl, K. Ota and N. Tokushige and put A = B = E to get lower bound (1).

Acknowledgements.
The work was supported by the Russian Scientific Foundation grant 16-11-10014. The author is grateful to A. Raigorodskii and F. Petrov for constant inspiration, to A. Kupavskii for historical review and for directing his attention to the paper [9] and to N. Rastegaev for very careful reading of the draft of the paper.