Tur\'an problems for simplicial complexes

TL;DR

This paper investigates extremal problems for simplicial complexes, establishing when their maximum edge counts are determined by hypergraph Turán numbers and identifying cases where this relation fails.

Contribution

It identifies classes of simplicial complexes with extremal numbers linked to hypergraph Turán numbers and highlights instances where this connection does not hold.

Findings

Certain simplicial complexes have extremal numbers determined by hypergraph Turán numbers.

The paper provides examples where the extremal number relation does not hold.

Progress on a problem posed by Conlon, Piga, and Schülke.

Abstract

An abstract simplicial complex $\mathbf{F}$ is a non-uniform hypergraph without isolated vertices, whose edge set is closed under taking subsets. The extremal number $\mathrm{ex}(n,\mathbf{F})$ is the maximum number of edges in an $\mathbf{F}$-free simplicial complex on $n$ vertices. This extremal number is naturally related to the generalised Tur\'an numbers of certain underlying hypergraphs. Making progress in a problem raised by Conlon, Piga, and Sch\"ulke, we find large classes of simplicial complexes whose extremal numbers are determined by the respective generalised hypergraph Tur\'an numbers. We also provide simplicial complexes for which such a relation does not hold.