Abstract

For a graph $F$, we say a hypergraph is a Berge-$F$ if it can be obtained from $F$ by replacing each edge of $F$ with a hyperedge containing it. A hypergraph is Berge-$F$-free if it does not contain a subhypergraph that is a Berge-$F$. The weight of a non-uniform hypergraph $\mathcal{H}$ is the quantity $\sum_{h \in E(\mathcal{H})} |h|$.
 Suppose $\mathcal{H}$ is a Berge-$F$-free hypergraph on $n$ vertices. In this short note, we prove that as long as every edge of $\mathcal{H}$ has size at least the Ramsey number of $F$, the weight of $\mathcal{H}$ is $o(n^2)$. This result is best possible in some sense. Along the way, we study other weight functions, and strengthen results of Gerbner and Palmer; and Grósz, Methuku and Tompkins.

Highlights

  • Generalizing the notion of hypergraph cycles due to Berge, the authors Gerbner and Palmer [6] introduced so-called Berge hypergraphs

  • The electronic journal of combinatorics 26(4) (2019), #P4.7 hypergraph H is Berge-F if there is an injection f : V (F ) → V (H) and bijection f : E(F ) → E(H) such that for every edge uv ∈ E(F ) we have {f (u), f (v)} ⊆ f

  • Note that for a fixed F there are many different hypergraphs that are Berge-F, and a fixed hypergraph H can be Berge-F for many different graphs F

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Summary

Introduction

Generalizing the notion of hypergraph cycles due to Berge, the authors Gerbner and Palmer [6] introduced so-called Berge hypergraphs. Combining Theorem 3 and Theorem 4, we can show the sum of the sizes of the edges (i.e., the weight) of a Berge-F -free hypergraph is o(n2) provided all the hyperedges are large enough, presenting another improvement of Theorem 1 and Theorem 2. This follows from a much more general theorem (which is presented below) by setting w(m) = m. If w(m) = Ω(m2), considering a single hyperedge of size n shows that the conclusion of Theorem 5 cannot hold This gives a Berge-F -free hypergraph H with h∈H w(|h|) w(n). For a hypergraph H, its 2-shadow is the graph whose edge-set is Γ(H) := ∪h∈HΓ({h}), i.e. all the edges contained in at least one hyperedge of H

Proof of Theorem 3
Proof of Theorem 4
Proof of Theorem 5

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