# 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.

## Key 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 $n \ge n_0(r, t)$.