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Ramsey properties of randomly perturbed hypergraphs

Elad Aigner-Horev, Dan Hefetz, Mathias Schacht

TL;DR

The paper resolves the Ramsey threshold for expanded 3-uniform cliques in randomly perturbed dense hypergraphs by introducing a novel tuple lemma for joint link graphs and a new hybrid hypergraph regularity framework that fuses strong and weak forms. The main result shows that adding a sparse random 3-uniform hypergraph with edge probability $p \u2192 n^{-1/M}$ (with $M=m_3( ilde K^{(3)}_t, ilde K^{(3)}_{t/2})$) to a dense seed yields a.a.s. two-color Ramsey behavior for $ ilde K^{(3)}_t$, with conjectured optimality for even $t$. The approach blends hypergraph regularity, a tuple-control mechanism for link structures, and Ramsey properties of the random component, with a detailed, multi-layered proof architecture including a new variant of the strong hypergraph regularity lemma. The results advance understanding of Ramsey phenomena under perturbation in hypergraphs and open pathways to further generalizations to higher uniformities and asymmetric configurations.

Abstract

We study Ramsey properties of randomly perturbed $3$-uniform hypergraphs. For~$t\geq 2$, write $\tilde K^{(3)}_t$ to denote the $3$-uniform {\it expanded} clique hypergraph obtained from the complete graph $K_t$ by expanding each of the edges of the latter with a new additional vertex. For an even integer $t\geq 4$, let~$M$ denote the asymmetric maximal density of the pair $(\tilde K^{(3)}_t,\tilde K^{(3)}_{t/2})$. We prove that adding a set~$F$ of random hyperedges satisfying $|F|\gg n^{3-1/M}$ to a given $n$-vertex $3$-uniform hypergraph~$H$ with non-vanishing edge density asymptotically almost surely results in a perturbed hypergraph enjoying the Ramsey property for $\tilde K^{(3)}_t$ and two colours. We conjecture that this result is asymptotically best possible with respect to the size of $F$ whenever $t\geq 6$ is even. The key tools of our proof are a new variant of the hypergraph regularity lemma accompanied with a \emph{tuple lemma} providing appropriate control over joint link graphs. Our variant combines the so called strong and the weak hypergraph regularity lemmata.

Ramsey properties of randomly perturbed hypergraphs

TL;DR

The paper resolves the Ramsey threshold for expanded 3-uniform cliques in randomly perturbed dense hypergraphs by introducing a novel tuple lemma for joint link graphs and a new hybrid hypergraph regularity framework that fuses strong and weak forms. The main result shows that adding a sparse random 3-uniform hypergraph with edge probability (with ) to a dense seed yields a.a.s. two-color Ramsey behavior for , with conjectured optimality for even . The approach blends hypergraph regularity, a tuple-control mechanism for link structures, and Ramsey properties of the random component, with a detailed, multi-layered proof architecture including a new variant of the strong hypergraph regularity lemma. The results advance understanding of Ramsey phenomena under perturbation in hypergraphs and open pathways to further generalizations to higher uniformities and asymmetric configurations.

Abstract

We study Ramsey properties of randomly perturbed -uniform hypergraphs. For~, write to denote the -uniform {\it expanded} clique hypergraph obtained from the complete graph by expanding each of the edges of the latter with a new additional vertex. For an even integer , let~ denote the asymmetric maximal density of the pair . We prove that adding a set~ of random hyperedges satisfying to a given -vertex -uniform hypergraph~ with non-vanishing edge density asymptotically almost surely results in a perturbed hypergraph enjoying the Ramsey property for and two colours. We conjecture that this result is asymptotically best possible with respect to the size of whenever is even. The key tools of our proof are a new variant of the hypergraph regularity lemma accompanied with a \emph{tuple lemma} providing appropriate control over joint link graphs. Our variant combines the so called strong and the weak hypergraph regularity lemmata.
Paper Structure (20 sections, 20 theorems, 158 equations, 1 figure)

This paper contains 20 sections, 20 theorems, 158 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Ramsey properties of random hypergraphs
  3. Main result
  4. A tuple lemma for link graphs
  5. A variant of the hypergraph regularity lemma
  6. Preliminaries
  7. Graph regularity
  8. Hypergraph regularity
  9. Monochromatic expanded cliques
  10. Properties of random hypergraphs
  11. Proof of Theorem \ref{['thm:main-Ramsey']}
  12. Proof of the tuple lemma
  13. A variant of the regularity lemma for hypergraphs
  14. Outline of the proof
  15. Proof of the regularity lemma
  16. ...and 5 more sections

Key Result

Theorem 1.1

For every $d > 0$ and every even integer $t \geq 4$, there exists a constant $C>0$ such that for every sequence of $3$-uniform $n$-vertex hypergraphs $(H_n)_{n\in\mathds N}$ with $e(H_n) \geq d n^3$ for every $n\in\mathds N$, we have whenever $p = p(n) \geq Cn^{- 1/M}$ for $M=m_3(\widetilde{K}^{(3)}_t, \widetilde{K}^{(3)}_{t/2})$.

Figures (1)

  • Figure 1: A single iteration of the refinement process

Theorems & Definitions (49)

  • Theorem 1.1: Main result
  • Proposition 1.2: Tuple lemma for joint links
  • Proposition 1.3: Variant of the regularity lemma for hypergraphs
  • Lemma 2.1: Triangle counting lemma
  • Lemma 2.2
  • proof
  • Lemma 2.3: Slicing lemma
  • Lemma 2.4: Tuple lemma for graphs
  • Lemma 2.5: Weak hypergraph regularity lemma
  • Lemma 2.6: Strong hypergraph regularity lemma
  • ...and 39 more