Uniform cross-$t$-intersecting families: proving Hirschorn's conjecture up to polynomial factor

TL;DR

This paper investigates the maximum product of sizes of two uniform cross-$t$-intersecting families, showing it is at most polynomially larger than Hirschorn's conjecture, but can sometimes be strictly bigger.

Contribution

It proves Hirschorn's conjecture up to a polynomial factor and highlights cases where the maximum exceeds the conjectured bound.

Findings

Maximum product is at most polynomially larger than Hirschorn's conjecture.

The maximum can be strictly bigger than the conjectured value.

Provides bounds and insights into the structure of cross-$t$-intersecting families.

Abstract

We consider a problem of maximizing the product of the sizes of two uniform cross-$t$-intersecting families of sets. We show that the value of this maximum is at most polynomially larger (in the size of a ground set) than a quantity conjectured by Hirschorn in 2008. At the same time, we observe that it can be strictly bigger.