A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix $F$, we say that a (0,1)-matrix $A$ has $F$ as a Berge hypergraph if there is a submatrix $B$ of $A$ and some row and column permutation of $F$, say $G$, with $G\le B$. Letting $\|A\|$ denote the number of columns in $A$, we define the extremal function $\mathrm{Bh}(m,{ F})=\max\{\|A\|\,:\, A \hbox{ }m\hbox{-rowed simple matrix and no Berge hypergraph }F\}$. We determine the asymptotics of $\mathrm{Bh}(m,F)$ for all $3$- and $4$-rowed $F$ and most $5$-rowed $F$. For certain $F$, this becomes the problem of determining the maximum number of copies of $K_r$ in a $m$-vertex graph that has no $K_{s,t}$ subgraph, a problem studied by Alon and Shikhelman.
@article{10_37236_6482,
author = {R. P. Anstee and Santiago Salazar},
title = {Forbidden {Berge} hypergraphs},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {1},
doi = {10.37236/6482},
zbl = {1358.05201},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6482/}
}
TY - JOUR
AU - R. P. Anstee
AU - Santiago Salazar
TI - Forbidden Berge hypergraphs
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/6482/
DO - 10.37236/6482
ID - 10_37236_6482
ER -
%0 Journal Article
%A R. P. Anstee
%A Santiago Salazar
%T Forbidden Berge hypergraphs
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/6482/
%R 10.37236/6482
%F 10_37236_6482
R. P. Anstee; Santiago Salazar. Forbidden Berge hypergraphs. The electronic journal of combinatorics, Tome 24 (2017) no. 1. doi: 10.37236/6482
