论文标题
布尔立方体中的章鱼：成对小交叉点的家庭，第一部分
Octopuses in the Boolean cube: families with pairwise small intersections, part I
论文作者
论文摘要
令$ \ mathcal f_1，\ ldots，\ Mathcal f_ \ ell $是$ \ {1，\ ldots，n \} $的子集的族。假设对于不同的$ k，k'$和任意$ f_1 \ in \ mathcal f_ {k {k}，f_2 \ in \ mathcal f_ {k'} $我们有$ | f_1 \ cap f_1 \ cap f_2 | \ le m。在这项工作中，我们发现该产品的渐近性是$ n $倾向于无限$ \ ell $和〜$ m $。 这个问题与Bohn等人的猜想有关。这是在2级多层理论中产生的，并要求在两级多层人士中提供最大的刻面和顶点的产物。该猜想最近由Weltge和第一作者解决。 主要结果可以根据着色来改写。我们对以下问题给出渐近答案。给定完整的$ M $均匀超图的边缘着色为$ \ ell $颜色，$ \ prod m_i $的最大值是什么，其中$ m_i $是$ i $ th颜色的单色cliques数量？
Let $\mathcal F_1, \ldots, \mathcal F_\ell$ be families of subsets of $\{1, \ldots, n\}$. Suppose that for distinct $k, k'$ and arbitrary $F_1 \in \mathcal F_{k}, F_2 \in \mathcal F_{k'}$ we have $|F_1 \cap F_2|\le m.$ What is the maximal value of $|\mathcal F_1|\ldots |\mathcal F_\ell|$? In this work we find the asymptotic of this product as $n$ tends to infinity for constant $\ell$ 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 $\ell$ colors, what is the maximum of $\prod M_i$, where $M_i$ is the number of monochromatic cliques in $i$-th color?
