On the maximum of the sum of the sizes of non-trivial cross-intersecting families

TL;DR

This paper investigates the maximum combined size of two non-trivial cross-intersecting families of subsets, introducing a strengthened shifting technique to solve the problem for uniform and non-uniform families.

Contribution

It provides a new bound for the sum of sizes of non-trivial cross-intersecting families and introduces a strengthened shifting method for the proof.

Findings

Determined the maximum of |F|+|G| for non-trivial cross-intersecting families.

Extended the results to families of different uniformities.

Introduced a strengthened shifting technique for combinatorial proofs.

Abstract

We consider families of k-subsets of the standard n-set. Two families F, G are said to be cross-intersecting if every member of F has non-empty intersection with every member of G. A family is called non-trivial if the intersection of all its members is empty. Supposing that F and G are non-trivial and cross-intersecting, we determine the maximum of |F|+|G|. For the proof a strengthened version of the so-called shifting technique is introduced. The corresponding problem for families of different uniformities is solved as well.