Octopuses in the Boolean cube: families with pairwise small intersections, part I
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Let ℱ_1, …, ℱ_ℓ be families of subsets of {1, …, n}. Suppose that for distinct k, k' and arbitrary F_1 ∈ℱ_k, F_2 ∈ℱ_k' we have |F_1 ∩ F_2|≤ m. What is the maximal value of |ℱ_1|… |ℱ_ℓ|? In this work we find the asymptotic of this product as n tends to infinity for constant ℓ and m. This question is related to a conjecture of Bohn et al. that arose in the 2-level polytope theory and asked for the largest product of the number of facets and vertices in a two-level polytope. This conjecture was recently resolved by Weltge and the first author. The main result can be rephrased in terms of colorings. We give an asymptotic answer to the following question. Given an edge coloring of a complete m-uniform hypergraph into ℓ colors, what is the maximum of ∏ M_i, where M_i is the number of monochromatic cliques in i-th color?
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