More complete intersection theorems

Abstract

The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of a k-uniform t-intersecting family on n points, and describes all optimal families. In recent work, we extended this theorem to the weighted setting, giving the maximum μp measure of a t-intersecting family on n points. In this work, we prove two new complete intersection theorems. The first gives the supremum μp measure of a t-intersecting family on infinitely many points, and the second gives the maximum cardinality of a subset of Zmn in which any two elements x,y have t positions i1,…,it such that xij−yij∈{−(s−1),…,s−1}. In both cases, we determine the extremal families, whenever possible.