Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ramsey lower bounds for bounded degree hypergraphs

Chunchao Fan, Qizhong Lin

Abstract

We prove that for all $k \ge 3$ and any integers $Δ, n$ with $n \ge 2^Δ,$ there exists a $k$-graph on $n$ vertices with maximum degree at most $Δ$ such that $r(H)\geq\tw_{k-1}(c_k Δ) \cdot n$ for some constant $c_k > 0$, where $\tw_k$ denotes the tower function. This makes the first progress toward a problem proposed by Conlon, Fox, and Sudakov (2009), who asked whether $r(H)\geq\tw_{k}(c_k Δ) \cdot n$ holds. Our proof relies on a novel construction of a $k$-graph on a growing number of vertices $n$ while keeping the maximum degree bounded by a fixed $Δ$.

Ramsey lower bounds for bounded degree hypergraphs

Abstract

We prove that for all and any integers with there exists a -graph on vertices with maximum degree at most such that for some constant , where denotes the tower function. This makes the first progress toward a problem proposed by Conlon, Fox, and Sudakov (2009), who asked whether holds. Our proof relies on a novel construction of a -graph on a growing number of vertices while keeping the maximum degree bounded by a fixed .

Paper Structure

This paper contains 5 sections, 7 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Stepping up and colorings
  3. Technical lemmas
  4. Proof of Theorem \ref{['main']}
  5. Concluding Remark

Key Result

Theorem 1.2

For any $k \ge 3$, there exists a constant $c_k> 0$ such that for any integers $\Delta \ge 1/c_k$ and $n\ge 2^{\Delta}$, there exists a $k$-uniform $n$-vertex hypergraph $H$ with maximum degree at most $\Delta$ such that

Theorems & Definitions (13)

  • Theorem 1.2
  • Lemma 3.1: B-H-S
  • Lemma 3.2: B-H-S
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.5
  • Definition 3.6
  • Lemma 3.7
  • Lemma 3.8
  • Definition 3.9
  • ...and 3 more