The Extremal Number of Surfaces. (11th May 2021)

• Record Type: Journal Article
• Authors: Kupavskii, Andrey; Polyanskii, Alexandr; Tomon, István; Zakharov, Dmitriy
• Is Part Of: International mathematics research notices. Volume 2022:Number 17(2022)
• Page Start: 13246
• Page End: 13271
• Publication Date: 2021-05-11
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab099
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: In 1973, Brown, Erdős and Sós proved that if $\mathcal{H}$ is a 3-uniform hypergraph on $n$ vertices which contains no triangulation of the sphere, then $\mathcal{H}$ has $O(n^{5/2})$ 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 $\mathcal{S}$ .