Simplicial Tur\'an problems

TL;DR

This paper introduces the study of extremal numbers in simplicial complexes, determining their asymptotic behavior for various natural complexes, including those with multiple incomplete layers.

Contribution

It systematically initiates the study of extremal problems in simplicial complexes and asymptotically determines extremal numbers for several natural cases.

Findings

Asymptotic determination of extremal numbers for various simplicial complexes

Identification of complexes with extremal examples having multiple incomplete layers

Establishment of foundational results for simplicial Turán problems

Abstract

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.