A hypergraph bandwidth theorem

Abstract: A cornerstone of extremal graph theory due to Erd\H{o}s and Stone states that the edge density which guarantees a fixed graph $F$ as subgraph also asymptotically guarantees a blow-up of $F$ as subgraph. It is natural to ask whether this phenomenon generalises to vertex-spanning structures such as Hamilton cycles. This was confirmed by B\"ottcher, Schacht and Taraz for graphs in the form of the Bandwidth Theorem. Our main result extends the phenomenon to hypergraphs. A graph on $n$ vertices that robustly contains a Hamilton cycle must satisfy certain conditions on space, connectivity and aperiodicity. Conversely, we show that if these properties are robustly satisfied, then all blow-ups of cycles on $n$ vertices with clusters of size at most $\operatorname{poly}(\log \log n)$ are guaranteed as subgraphs. This generalises to powers of cycles and to the hypergraph setting. As an application, we recover a series of classic results and recent breakthroughs on Hamiltonicity under degree conditions, which are then immediately upgraded to blown up versions. The proofs are based on a new setup for embedding large substructures into dense hypergraphs, which is of independent interest and does not rely on the Regularity Lemma or the Absorption Method.