The extremal number of Venn diagrams

TL;DR

This paper proves that any sufficiently large family of binary vectors must contain three sets forming a Venn diagram with all eight regions non-empty, establishing a sharp bound on the extremal number related to Venn diagrams.

Contribution

It improves the known bound on the size threshold for families to guarantee a Venn diagram with all regions non-empty, sharpening previous results.

Findings

Established a bound of Cn^3 for the family size to contain a 3-set Venn diagram.

Improved previous bound of Cn^{3.75} by Gupta, Lee, and Li.

Result is sharp up to the constant factor.

Abstract

We show that there exists an absolute constant $C>0$ such that any family $\mathcal{F}\subset \{0,1\}^n$ of size at least $Cn^3$ has dual VC-dimension at least 3. Equivalently, every family of size at least $Cn^3$ contains three sets such that all eight regions of their Venn diagram are non-empty. This improves upon the $Cn^{3.75}$ bound of Gupta, Lee and Li and is sharp up to the value of the constant.