Traces Without Maximal Chains

Abstract

The trace of a family of sets ${\cal A}$ on a set $X$ is ${\cal A}|_X=\{A\cap X:A\in {\cal A}\}$. If ${\cal A}$ is a family of $k$-sets from an $n$-set such that for any $r$-subset $X$ the trace ${\cal A}|_X$ does not contain a maximal chain, then how large can ${\cal A}$ be? Patkós conjectured that, for $n$ sufficiently large, the size of ${\cal A}$ is at most ${n-k+r-1\choose r-1}$. Our aim in this paper is to prove this conjecture.