Tiling dense hypergraphs

TL;DR

This paper establishes conditions under which perfect tilings in hypergraphs are asymptotically guaranteed, extending known results to broader hereditary hypergraph families including those defined by degree and quasirandomness.

Contribution

It proves that the three necessary conditions for perfect tilings are also sufficient for all hereditary hypergraph families, broadening the scope of tiling theory.

Findings

Confirmed sufficiency of conditions for hereditary hypergraph families

Extended classical tiling results to new hypergraph classes

Unified various tiling results under a common framework

Abstract

In the perfect tiling problem, we aim to cover the vertices of a hypergraph~$G$ with pairwise vertex-disjoint copies of a hypergraph $F$. There are three essentially necessary conditions for such a perfect tiling, which correspond to barriers in space, divisibility and covering. It is natural to ask in which situations these conditions are also asymptotically sufficient. Our main result confirms this for all hypergraph families that are hereditary in the sense of being approximately closed under taking typical induced subgraphs of constant order. Among others, this includes families parametrised by minimum degrees and quasirandomness, which have been studied extensively in this area. As an application, we recover and extend a series of well-known results for perfect tilings in hypergraphs and related settings involving vertex orderings and rainbow structures.