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A hypergraph bandwidth theorem

Richard Lang, Nicolás Sanhueza-Matamala

Abstract

A cornerstone of extremal graph theory due to Erdős 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 Böttcher, 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.

A hypergraph bandwidth theorem

Abstract

A cornerstone of extremal graph theory due to Erdős and Stone states that the edge density which guarantees a fixed graph as subgraph also asymptotically guarantees a blow-up of as subgraph. It is natural to ask whether this phenomenon generalises to vertex-spanning structures such as Hamilton cycles. This was confirmed by Böttcher, Schacht and Taraz for graphs in the form of the Bandwidth Theorem. Our main result extends the phenomenon to hypergraphs. A graph on 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 vertices with clusters of size at most 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.

Paper Structure

This paper contains 55 sections, 69 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and applications
  3. A framework for Hamiltonicity
  4. Necessary conditions
  5. Robustness
  6. Sufficient conditions
  7. Methodology
  8. Notation
  9. Proofs of the applications
  10. Inheritance
  11. Vertex-degree in $3$-uniform graphs
  12. Power of Hamilton cycles
  13. Natural frameworks
  14. Singleton vicinities
  15. Pair vicinities
  16. ...and 40 more sections

Key Result

Theorem 1.1

For every $1 \leq d < k \leq t$ and $\varepsilon >0$, there exist $c$ and $n_0$ with the following properties. Let $G$ be a $k$-uniform hypergraph on $n\geq n_0$ vertices with Let $H$ be an $n$-vertex blow-up of the $(t-k+1)$st power of a $k$-uniform cycle with clusters of (not necessarily uniform) size at most $(\log \log n)^c$. Then $H \subseteq G$.

Figures (1)

  • Figure 1: A blow-up cover whose shape is the path $uvw$. The blow-ups $R^u(\mathcal{V}^u)$, $R^v(\mathcal{V}^v)$ and $R^w(\mathcal{V}^w)$ are coloured green. The blow-ups $R^{uv}(\mathcal{W}^{uv})$ and $R^{vw}(\mathcal{W}^{vw})$ are coloured red and blue, respectively. The filled nodes present the singleton clusters. The $2$-graph $R^v$ has vertices $1,2,3,4,5$ and edges $12,23,34,14,15,25$. The corresponding set family is $\mathcal{V}^v=\{V_1,V_2,V_3,V_4,V_5\}$ with exceptional cluster $V_5$. The set family $\mathcal{V}^{v} \smallsetminus \{V_5\}$ is hit by the family $\mathcal{W}^{uv}_v = \{W_1,W_2,W_3,W_4\} \subseteq \mathcal{W}^{uv}$.

Theorems & Definitions (150)

  • Theorem 1.1: Hypergraph Bandwidth Theorem
  • Theorem 2.1: Bandwidth Theorem
  • Conjecture 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6: BR20PSS23
  • Conjecture 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 140 more