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A lower bound on the Ramsey number $R_k(k+1,k+1)$

Pavel Pudlák, Vojtěch Rödl, William J. Wesley

Abstract

We will prove that $R_k(k+1,k+1)\geq 4 tw_{\lfloor k/4\rfloor -3}(2)$, where $tw$ is the tower function defined by ${tw}_1(x)=x$ and ${tw}_{i+1}(x)=2^{{tw}_i(x)}$. We also give proofs of $R_k(k+1,k+2)\geq 4 tw_{k-7}(2)$, $R_k(k+1,2k+1)\geq 4 tw_{k-3}(2)$, and $R_k(k+2,k+2)\geq 4 tw_{k-4}(2)$.

A lower bound on the Ramsey number $R_k(k+1,k+1)$

Abstract

We will prove that , where is the tower function defined by and . We also give proofs of , , and .

Paper Structure

This paper contains 8 sections, 4 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The lower bound $R_k(k+1,k+1)\geq s(\lfloor k/4\rfloor)$
  4. The way to the lower bound on $R_k(k+1,k+1)$.
  5. Conclusions
  6. Proof of $P_k(2,3)\geq s(k-4)$.
  7. Proof of $P_k(3,3)\geq s(k-1)$.
  8. Proof of $R_k(k+1,2k+1)\geq s(k)$.

Key Result

Theorem 2.1

For every $k\geq 4$, $R_k(k+1,k+1)\geq s(\lfloor k/4\rfloor)$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (9)

  • Theorem 2.1
  • Definition 3.1: bridge
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Definition 4.1: $n$-bridge
  • Claim 4.1
  • Proposition A.1