Traces without maximal chains (Q2380427)

Summary: 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? Patkós 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.