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The Ramsey number for 4-uniform tight cycles

Allan Lo, Vincent Pfenninger

Abstract

A $k$-uniform tight cycle is a $k$-graph with a cyclic ordering of its vertices such that its edges are precisely the sets of $k$ consecutive vertices in that ordering. A $k$-uniform tight path is a $k$-graph obtained by deleting a vertex from a $k$-uniform tight cycle. We prove that the Ramsey number for the $4$-uniform tight cycle on $4n$ vertices is $(5 +o(1))n$. This is asymptotically tight. This result also implies that the Ramsey number for the $4$-uniform tight path on $n$ vertices is $(5/4 + o(1))n$.

The Ramsey number for 4-uniform tight cycles

Abstract

A -uniform tight cycle is a -graph with a cyclic ordering of its vertices such that its edges are precisely the sets of consecutive vertices in that ordering. A -uniform tight path is a -graph obtained by deleting a vertex from a -uniform tight cycle. We prove that the Ramsey number for the -uniform tight cycle on vertices is . This is asymptotically tight. This result also implies that the Ramsey number for the -uniform tight path on vertices is .

Paper Structure

This paper contains 10 sections, 18 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Sketch of the proof of Theorem \ref{['thm:main']}
  3. Organisation of the Paper
  4. Preliminaries
  5. Blow-ups
  6. Blueprints
  7. Finding monochromatic tightly connected matchings
  8. Proof of Lemma 5.1 assuming Lemma 5.3
  9. Sketch of the proof of Lemma \ref{['lem:increase_matching']} and suitable pairs
  10. Proof of Lemma 4.4 assuming (H1)

Key Result

Theorem 1.1

Let $\varepsilon > 0$. For $n$ large enough we have $r(C_{4n}^{(4)}) \leq (5 + \varepsilon)n$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Proposition 1.2
  • proof
  • Corollary 1.3
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 4.1
  • ...and 35 more