Anticlusters and Intersecting Families of Sets and \(t\)-valued Functions

It is shown that if \([n] = X_1 \cup X_2 \cup \cdots \cup X_l\) is a partition of \([n]\) and if \(S_t\) is a family of \(t\)-valued functions intersecting on at least one element of \(k\) (circularly) consecutive blocks, then \(|S_t| < t^{n-k}\). If given \(a_1 < a_2 < \cdots < a_y \leq l \), \(\acute{S}_t\) is a family of \(t\)-valued functions intersecting on at least one element of \(X_{a_{1}+m}, X_{a_{2}+m}, \ldots, X_{a_{k}+m}\) for some \(m\) with \(1-a_1 \leq m \leq n – a_k\), then \(|\acute{S}_t| \leq t^{n-k}\). Both these results were conjectured by Faudree, Schelp, and Sós [FSS]. The main idea of our proofs is that of anticlusters introduced by Griggs and Walker [GW] which we discuss in some detail. We also discuss several related intersection theorems about sets, \(2\)-valued functions, and \(t\)-valued functions.