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.
Sean English
1
;
Dániel Gerbner
2
;
Abhishek Methuku
3
;
Cory Palmer
4
1
Ryerson University, Toronto
2
Alfred Renyi Institute of Mathematics
3
École Polytechnique Fédérale de Lausanne,
4
University of Montana, Missoula
Sean English; Dániel Gerbner; Abhishek Methuku; Cory Palmer. On the weight of Berge-\(F\)-free hypergraphs. The electronic journal of combinatorics, Tome 26 (2019) no. 4. doi: 10.37236/8504
@article{10_37236_8504,
author = {Sean English and D\'aniel Gerbner and Abhishek Methuku and Cory Palmer},
title = {On the weight of {Berge-\(F\)-free} hypergraphs},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {4},
doi = {10.37236/8504},
zbl = {1422.05073},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8504/}
}
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AU - Dániel Gerbner
AU - Abhishek Methuku
AU - Cory Palmer
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JO - The electronic journal of combinatorics
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VL - 26
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