A note on the maximum size of Berge--free hypergraphs
Introduction
A Berge cycle of length , denoted by Berge-, is an alternating sequence of distinct vertices and distinct hyperedges of the form where for each and .
Throughout the paper we allow hypergraphs to include multiple copies of the same hyperedge (multi-hyperedges).
Let be a Berge--free hypergraph on vertices. Győri and Lemons [3] showed that . Notice that it is natural to take in the sum, otherwise we could have arbitrarily many copies of a -vertex hyperedge. In [2] Gerbner and Palmer improved the upper bound proving that , furthermore they showed that there exists a Berge--free hypergraph such that .
In this paper we improve their bounds.
Theorem 1 Let be a Berge--free hypergraph on vertices, then Furthermore, there exists a -free hypergraph on vertices, such that
This improves the upper-bound by factor of and slightly increases the lower-bound.
Note that Theorem 1 provides an upper bound on the maximum number of edges in Berge--free -uniform hypergraphs. Namely, for , if is an -uniform Berge--free hypergraph, then .
We introduce couple of important notations and definitions used throughout the paper.
-path (-walk) is a path (walk) consisting of vertices. For convenience, an edge or a pair of vertices is sometimes referred to as . For a graph (or a hypergraph) , for convenience, we sometimes use to denote the edge set of the graph (hypergraph) . Thus the number of edges (hyperedges) in is .
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 on a vertex set by embedding edges into each hyperedge of . More specifically, for each we embed edges on the vertices of , such that collection of edges that were embedded in consists of pairwise vertex-disjoint triangles and edges (each edge is allowed to be embedded in at most one hyperedge). We say that has color if was embedded in the hyperedge 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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