The Tur\'an Number of Surfaces

TL;DR

This paper establishes a threshold for the number of edges in 3-uniform hypergraphs that guarantees the presence of a triangulation of the real projective plane, thereby resolving a conjecture and determining Turán numbers for all surfaces.

Contribution

It proves a new extremal bound for hypergraphs containing surface triangulations, resolving a key conjecture and advancing the understanding of Turán numbers for surfaces.

Findings

Existence of a constant c such that hypergraphs with at least c n^{5/2} edges contain a projective plane triangulation.

Resolution of a conjecture by Kupavskii et al. regarding surface embeddings in hypergraphs.

Asymptotic determination of Turán numbers for all surfaces.

Abstract

We show that there is a constant $c$ such that any 3-uniform hypergraph $\mathcal H$ with $n$ vertices and at least $cn^{5/2}$ edges contains a triangulation of the real projective plane as a subgraph. This resolves a conjecture of Kupavskii, Polyanskii, Tomon, and Zakharov. Furthermore, our work, combined with prior results, asymptotically determines the Tur\'an number of all surfaces.