On the sizes of \(t\)-intersecting \(k\)-chain-free families
Abstract
A set system \({\mathcal F}\) is \(t\)-intersecting, if the size of the intersection of every pair of its elements has size at least \(t\). A set system \({\mathcal F}\) is \(k\)-Sperner, if it does not contain a chain of length \(k+1\). Our main result is the following: Suppose that \(k\) and \(t\) are fixed positive integers, where \(n+t\) is even and \(n\) is large enough. If \({\mathcal F}\subseteq 2^{[n]}\) is a \(t\)-intersecting \(k\)-Sperner family, then \(|{\mathcal F}|\) has size at most the size of the sum of \(k\) layers, of sizes \((n+t)/2,\ldots, (n+t)/2+k-1\). This bound is best possible. The case when \(n+t\) is odd remains open.
Mathematics Subject Classifications: 05D05
Keywords: Extremal set theory, Sperner families, intersection theorems