Decompositions of quasirandom hypergraphs into hypergraphs of bounded degree

TL;DR

This paper proves that quasirandom hypergraphs can be approximately decomposed into bounded degree hypergraphs, extending to multipartite and sparse cases, with implications for hypergraph and simplicial complex decompositions.

Contribution

It introduces new approximate decomposition methods for quasirandom hypergraphs into bounded degree hypergraphs, including multipartite and sparse settings.

Findings

Decomposition of quasirandom hypergraphs into bounded degree hypergraphs.

Applicability to multipartite and sparse hypergraph settings.

Implications for hypergraph and simplicial complex decompositions.

Abstract

We prove that any quasirandom uniform hypergraph $H$ can be approximately decomposed into any collection of bounded degree hypergraphs with almost as many edges. In fact, our results also apply to multipartite hypergraphs and even to the sparse setting when the density of $H$ quickly tends to $0$ in terms of the number of vertices of $H$. Our results answer and address questions of Kim, K\"uhn, Osthus and Tyomkyn; and Glock, K\"uhn and Osthus as well as Keevash. The provided approximate decompositions exhibit strong quasirandom properties which is very useful for forthcoming applications. Our results also imply approximate solutions to natural hypergraph versions of long-standing graph decomposition problems, as well as several decomposition results for (quasi)random simplicial complexes into various more elementary simplicial complexes such as triangulations of spheres and other manifolds.