The extremal number of surfaces

TL;DR

This paper extends classical extremal results on 3-uniform hypergraphs avoiding certain surface triangulations from spheres to tori and all closed orientable surfaces, establishing optimal bounds on the number of edges.

Contribution

It proves that the maximum number of edges in hypergraphs avoiding triangulations of tori and other surfaces matches the known bounds for spheres, resolving a conjecture and generalizing previous results.

Findings

Maximum edges in hypergraphs avoiding surface triangulations established

Results apply to all closed orientable surfaces

Confirms conjecture for tori and extends to general surfaces

Abstract

In 1973, Brown, Erd\H{o}s and S\'os proved that if $\mathcal{H}$ is a 3-uniform hypergraph on $n$ vertices which contains no triangulation of the sphere, then $\mathcal{H}$ has at most $O(n^{5/2})$ edges, and this bound is the best possible up to a constant factor. Resolving a conjecture of Linial, also reiterated by Keevash, Long, Narayanan, and Scott, we show that the same result holds for triangulations of the torus. Furthermore, we extend our result to every closed orientable surface $\mathcal{S}$.