Minimum degree and diversity in intersecting antichains

Abstract

Let $n, \ell$ be positive integers, $n > 2\ell + 1$. Let X be an n-element set and $\mathcal{F}$ an antichain, $\mathcal{F} \subset 2^X$. Kiselev, Kupavskii and Patkós conjectured that if $|F\cup G| \leq 2\ell + 1$ for all $F, G \in \mathcal{F}$ then the minimum degree of $\mathcal{F}$ is no more than ${n - 1\choose \ell - 1}$, the minimum degree of ${[n]\choose \ell}$. We prove this conjecture for $n \geq \ell^3 + \ell^2 + \frac32 \ell$ and solve the analogous problem for the case $|F \cap G| \geq t$.