Set systems without a simplex, Helly hypergraphs and union-efficient families

TL;DR

This paper explores set families with specific intersection properties, providing new bounds and solutions for related combinatorial problems using advanced inequalities and graph theory techniques.

Contribution

It solves a boundary case of Tuza's problem for non-trivial q-Helly families by applying Karamata's inequality and analyzing 2-self-centered graphs.

Findings

Determined the minimum size of 2-self-centered graphs with clique conditions

Solved a boundary case of Tuza's problem for q-Helly families

Provided equivalent formulations for set families with bounded intersection properties

Abstract

We present equivalent formulations for concepts related to set families for which every subfamily with empty intersection has a bounded sub-collection with empty intersection. Hereby, we summarize the progress on the related questions about the maximum size of such families. In this work we solve a boundary case of a problem of Tuza for non-trivial $q$-Helly families, by applying Karamata's inequality and determining the minimum size of a $2$-self-centered graph for which the common neighborhood of every pair of vertices contains a clique of size $q-2$.