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Simplicial complex Mathematics Combinatorics h-vector Abstract simplicial complex Simplicial homology Simplicial approximation theorem Context (archaeology) Simplicial manifold Set (abstract data type) Simplicial set Discrete mathematics Pure mathematics Computer science Geography Programming language Homotopy category Archaeology Homotopy

David Conlon , Simón Piga , Bjarne Schülke ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2310.01822 · OA: W4387355907

A simplicial complex $H$ consists of a pair of sets $(V,E)$ where $V$ is a set of vertices and $E\subseteq\mathscr{P}(V)$ is a collection of subsets of $V$ closed under taking subsets. Given a simplicial complex $F$ and $n\in \mathbb N$, the extremal number $\text{ex}(n,F)$ is the maximum number of edges that a simplicial complex on $n$ vertices can have without containing a copy of $F$. We initiate the systematic study of extremal numbers in this context by asymptotically determining the extremal numbers of several natural simplicial complexes. In particular, we asymptotically determine the extremal number of a simplicial complex for which the extremal example has more than one incomplete layer.