The Cameron-Liebler problem for sets

TL;DR

This paper introduces and completely solves the Cameron-Liebler problem for sets, establishing links with Erdős-Ko-Rado results and providing a characterization theorem for these classes.

Contribution

It extends the Cameron-Liebler framework to sets, offering a complete solution and a new characterization theorem, connecting it with classical combinatorial results.

Findings

Complete solution to the Cameron-Liebler problem for sets

Characterization theorem for Cameron-Liebler classes of sets

Established links with Erdős-Ko-Rado results

Abstract

Cameron-Liebler line classes and Cameron-Liebler k-classes in PG(2k+1,q) are currently receiving a lot of attention. Links with the Erd\H{o}s-Ko-Rado results in finite projective spaces occurred. We introduce here in this article the similar problem on Cameron-Liebler classes of sets, and solve this problem completely, by making links to the classical Erd\H{o}s-Ko-Rado result on sets. We also present a characterisation theorem for the Cameron-Liebler classes of sets.