Families intersecting on an interval

TL;DR

This paper investigates a combinatorial problem related to intersecting families of subsets of [n], focusing on the maximum size of such families where intersections contain cyclic translates of a fixed subset B, providing a new proof of existing results.

Contribution

It offers an alternative, more direct proof of a known Erdős–Ko–Rado-type theorem for families intersecting on cyclic translates of a subset B.

Findings

Maximum family size is achieved by all supersets of B when B is a block of length t.

Provides a more direct proof of the existing theorem.

Confirms the optimality of the supersets of B as the largest intersecting family.

Abstract

We shall be interested in the following Erdos-Ko-Rado-type question. Fix some subset B of [n]. How large a family A of subsets of [n] can we find such that the intersection of any two sets in A contains a cyclic translate (modulo n) of B? Chung, Graham, Frankl and Shearer have proved that, in the case where B is a block of length t, we can do no better than to take A to consist of all supersets of B. We give an alternative proof of this result, which is in a certain sense more 'direct'.