A note on maximum size of Berge-$C_4$-free hypergraphs

TL;DR

This paper investigates the maximum total size of hyperedges in hypergraphs that do not contain a Berge 4-cycle, improving existing bounds through new theoretical results and constructions.

Contribution

It significantly improves the upper bound and slightly enhances the lower bound for the maximum sum of hyperedge sizes in Berge-4-cycle-free hypergraphs.

Findings

Improved upper bound for hypergraph size without Berge 4-cycle

New construction slightly improves the lower bound

Advances understanding of extremal hypergraph properties

Abstract

In this paper, we consider maximum possible value for the sum of cardinalities of hyperedges of a hypergraph without a Berge $4$-cycle. We significantly improve the previous upper bound provided by Gerbner and Palmer. Furthermore, we provide a construction that slightly improves the previous lower bound.