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05.12.2022 | 15:00
Thema der Dissertation:
Minimal Ramsey graphs, orthogonal Latin squares, and hyperplane coverings
Thema der Disputation:
The extremal number of surfaces
Abstract: How many edges can an $n$-vertex graph have if it does not contain a fixed graph $H$ as a subgraph? This is a classic problem in extremal graph theory, dating back to the work of Mantel in the early 20th century. The answer is known as the \emph{extremal number} of $H$. The study of extremal numbers of graphs, and subsequently also of hypergraphs, has received considerable attention from researchers over the past century. In this talk, we will explore this extremal problem in the setting of 3-uniform hypergraphs, forbidding a \emph{family} of topologically defined hypergraphs.
A 3-uniform hypergraph $H$ can be viewed as a simplicial complex whose faces are given by all subsets $X\subseteq V(H)$ such that $X$ is fully contained in some hyperedge of $H$. We say that $H$ is \emph{homeomorphic} to a given topological space $T$ if the simplicial complex associated to $H$ is homeomorphic to $T$. In the 1970s, Brown, Erd\H{o}s, and S\'os studied the largest number of edges in an $n$-vertex 3-uniform hypergraph not containing a subhypergraph homeomorphic to the 2-sphere, and determined the answer up to a constant factor. A natural next step is to consider the corresponding question for other surfaces. We will present two recent results, due to Kupavskii, Polyanskii, Tomon, and Zakharov and Sankar, resolving this problem for any closed surface up to a constant factor.

### Zeit & Ort

05.12.2022 | 15:00

Seminarraum 032
(Fachbereich Mathematik und Informatik, Arnimallee 6, 14195 Berlin)