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Tuesday, July 7, 2020

2:00 pm in Zoom,Tuesday, July 7, 2020

A survey of Berge-Turan hypergraph problems

Cory Palmer (University of Montana)

Abstract: For a graph $F$, we say that a hypergraph $\mathcal{H}$ is a Berge-$F$ if there is an injection $f: V(F) \rightarrow V(\mathcal{H})$ and bijection $f':E(F)\rightarrow E(\mathcal{H})$ such that for every edge $uv\in E(F)$ we have $\{f(u),f(v)\}\subseteq f'(uv)$. Alternatively, $\mathcal{H}$ is Berge-$F$ if we can embed a distinct graph edge into each hyperedge of $\mathcal{H}$ to obtain a copy of $F$. Note that for a fixed $F$ there are many different hypergraphs that are a Berge-$F$ and a fixed hypergraph $\mathcal{H}$ can be a Berge-$F$ for more than one graph $F$.

A hypergraph is Berge-$F$-free if it contains no subhypergraph isomorphic to any Berge-$F$. The maximum number of edges in an Berge-$F$-free $n$-vertex $r$-graph is denoted $\mathrm{ex}_r(n,\textrm{Berge-}F)$. Observe that when $r=2$, then a Berge-$F$ is simply the graph $F$ and then we are investigating the classical Turan function $\mathrm{ex}(n,F)$.

Early work on $\mathrm{ex}_r(n,\textrm{Berge-}F)$ focused on the case when $F$ is a path or a cycle. Results of Gyori, Katona, and Lemons and Davoodi, Gyori, Methuku and Tompkins establish an analogue of the Erdos-Gallai theorem for Berge paths. Gyori and Lemons proved $\mathrm{ex}_r(n,\textrm{Berge-}C_{2k}) = O(n^{1+1/k})$ for $r \geq 3$. This matches the order of magnitude of the bound found in the graph case. They prove the same upper bound for $\textrm{Berge-}C_{2k+1}$-free hypergraphs which is significantly different from the graph case.

In this talk we will survey results on the function $\mathrm{ex}_r(n,\textrm{Berge-}F)$ for various graphs $F$. We will also discuss the connection to other extremal problems and give a number of interesting open problems.

Please contact Sean at SEnglish (at) illinois (dot) edu for the Zoom information.