Incompatible intersection properties

TL;DR

This paper investigates the maximum size of a family of sets with specific intersection properties, proving an upper bound close to the theoretical maximum and proposing a conjecture to improve this bound.

Contribution

It establishes a nearly optimal upper bound for families with certain intersection conditions and introduces a conjecture to further tighten this bound.

Findings

Proved that such a family has size at most 2^{n-2}.

Identified the bound as nearly best possible under current conditions.

Proposed a conjecture that could reduce the intersection requirement from 38 to 3.

Abstract

Let $\mathcal F\subset 2^{[n]}$ be a family in which any three sets have non-empty intersection and any two sets have at least $38$ elements in common. The nearly best possible bound $|\mathcal F|\le 2^{n-2}$ is proved. We believe that $38$ can be replaced by $3$ and provide a simple-looking conjecture that would imply this.