Disjoint pairs in set systems with restricted intersection

TL;DR

This paper establishes bounds on the maximum number of disjoint pairs in set systems with intersection restrictions, advancing understanding in extremal combinatorics and related intersection problems.

Contribution

It provides the first asymptotically optimal upper bounds on disjoint pairs in set systems avoiding certain cross-intersections, including uniform families.

Findings

Upper bound on disjoint pairs in cross-intersecting set systems.

Asymptotic bounds for disjoint pairs in single $t$-avoiding families.

Connection between disjoint pairs and maximum product in uniform families.

Abstract

The problem of bounding the size of a set system under various intersection restrictions has a central place in extremal combinatorics. We investigate the maximum number of disjoint pairs a set system can have in this setting. In particular, we show that for any pair of set systems $(\mathcal{A}, \mathcal{B})$ which avoid a cross-intersection of size $t$, the number of disjoint pairs $(A, B)$ with $A \in \mathcal{A}$ and $B \in \mathcal{B}$ is at most $\sum_{k=0}^{t-1}\binom{n}{k}2^{n-k}$. This implies an asymptotically best possible upper bound on the number of disjoint pairs in a single $t$-avoiding family $\mathcal{F} \subset \mathcal{P}[n]$. We also study this problem when $\mathcal{A}$, $\mathcal{B} \subset [n]^{(r)}$ are both $r$-uniform, and show that it is closely related to the problem of determining the maximum of the product $|\mathcal{A}||\mathcal{B}|$ when $\mathcal{A}$ and $\mathcal{B}$ avoid a cross-intersection of size $t$, and $n \ge n_0(r, t)$.

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