Two extremal problems on intersecting families

Abstract

In this short note, we address two problems in extremal set theory regarding intersecting families. The first problem is a question posed by Kupavskii: is it true that given two disjoint cross-intersecting families $\mathcal{A},\mathcal{B}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$, they must satisfy $\min \{|\mathcal{A}|,|\mathcal{B}|\}\le \frac{1}{2}\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)$? We give an affirmative answer for $n\ge 2k^2$, and construct families showing that this range is essentially the best one could hope for, up to a constant factor. The second problem is a conjecture of Frankl. It states that for $n\ge 3k$, the maximum diversity of an intersecting family $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ is equal to $\left(\genfrac{}{}{0ex}{}{n-3}{k-2}\right)$. We are able to find a construction beating the conjectured bound for $n$ slightly larger than $3k$, which also disproves a conjecture of Kupavskii.

Cited by in corpus (1)

• Structure and properties of large intersecting families