The maximum sum of the sizes of all intersections within $m$-size families

TL;DR

This paper determines the maximum sum of intersections within a family of sets of fixed size, showing that lexicographically ordered families maximize this sum for large n, extending classical combinatorial results.

Contribution

It proves that lexicographical initial segments maximize the sum of pairwise intersections in large set families, extending previous work on adjacent pairs and graph edges.

Findings

Lexicographical families maximize the sum of intersections.

Provides a sharp upper bound for the sum of degrees squared in hypergraphs.

Extends classical combinatorial theorems to intersection sums.

Abstract

For a family of sets $\mathcal{F}$, let $\omega(\mathcal{F}):=\sum_{\{A,B\}\subset \mathcal{F}}|A\cap B|$. In this paper, we prove that provided $n$ is sufficiently large, for any $\mathcal{F}\subset \binom{[n]}{k}$ with $|\mathcal{F}|=m$, $\omega(\mathcal{F})$ is maximized by the family consisting of the first $m$ sets in the lexicographical ordering on $\binom{[n]}{k}$. Compared to the maximum number of adjacent pairs in families, determined by Das, Gan and Sudakov in 2016, $\omega(\mathcal{F})$ distinguishes the contributions of intersections of different sizes. Then our results is an extension of Ahlswede and Katona's results in 1978, which determine the maximum number of adjacent edges in graphs. Besides, since $\omega(\mathcal{F})=\frac{1}{2}\left(\sum_{x\in [n]}|\{F\in \mathcal{F}:x\in F\}|^2-km\right)$ for $k$-uniform family with size $m$, our results also give a sharp upper bound of the sum of squares of degrees in a hypergraph.