A note on the maximum size of Berge-C4-free hypergraphs

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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.

Introduction

A Berge cycle of length k, denoted by Berge-Ck, is an alternating sequence of distinct vertices and distinct hyperedges of the form v1,h1,v2,h2,,vk,hk where vi,vi+1hi for each i{1,2,,k1} and vk,v1hk.

Throughout the paper we allow hypergraphs to include multiple copies of the same hyperedge (multi-hyperedges).

Let H be a Berge-C4-free hypergraph on n vertices. Győri and Lemons [3] showed that hH(|h|3)(1+o(1))122n32. Notice that it is natural to take |h|3 in the sum, otherwise we could have arbitrarily many copies of a 3-vertex hyperedge. In [2] Gerbner and Palmer improved the upper bound proving that hH(|h|3)62n32+O(n), furthermore they showed that there exists a Berge-C4-free hypergraph H such that hH(|h|3)(1+o(1))133n32.

In this paper we improve their bounds.

Theorem 1

Let H be a Berge-C4-free hypergraph on n vertices, then hH(|h|3)(1+o(1))12n32.

Furthermore, there exists a C4-free hypergraph H on n vertices, such that (1+o(1))126n32hH(|h|3).

This improves the upper-bound by factor of 6 and slightly increases the lower-bound.

Note that Theorem 1 provides an upper bound on the maximum number of edges in Berge-C4-free r-uniform hypergraphs. Namely, for r4, if H is an r-uniform Berge-C4-free hypergraph, then |E(H)|12(r3)n32+o(n32).

We introduce couple of important notations and definitions used throughout the paper.

3-path (3-walk) is a path (walk) consisting of 3 vertices. For convenience, an edge or a pair of vertices {a,b} is sometimes referred to as ab. For a graph (or a hypergraph) H, for convenience, we sometimes use H to denote the edge set of the graph (hypergraph) H. Thus the number of edges (hyperedges) in H is |H|.

Section snippets

Proof of Theorem 1

We will now construct a graph, the existence of which is proved in [2] (page 10). Let us take a graph H on a vertex set V(H) by embedding edges into each hyperedge of H. More specifically, for each hH we embed |h|3 edges on the vertices of h, such that collection of edges that were embedded in h consists of pairwise vertex-disjoint triangles and edges (each edge is allowed to be embedded in at most one hyperedge). We say that eH has color h if e was embedded in the hyperedge h of the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

I want to thank my colleagues Abhishek Methuku and Ervin Győri for having extremely helpful discussions about this problem. I thank the anonymous referees for their invaluable comments and suggestions.

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