INTERSECTING FAMILIES OF SEPARATED SETS

Abstract

A set $A\subseteq \{1,2,\ldots,n\}$ is said to be $k$-separated if, when considered on the circle, any two elements of $A$ are separated by a gap of size at least $k$.

A conjecture due to Holroyd and Johnson that an analogue of the Erdős–Ko–Rado theorem holds for $k$-separated sets is proved. In particular, the result holds for the vertex-critical subgraph of the Kneser graph identified by Schrijver, the collection of separated sets. A version of the Erdős–Ko–Rado theorem for weighted $k$-separated sets is also given.