Traces Without Maximal Chains

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