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

Pavel Pudlák; Vojtěch Rödl; William J. Wesley

doi:10.19086/aic.2026.3

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

Combinatorics 2026-04-27 v4

Authors: Pavel Pudlák , Vojtěch Rödl , William J. Wesley

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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 $R_k(k+2,k+2)\geq 4 tw_{k-4}(2)$ .

Keywords

Ramsey number tower function lower bound graph theory combinatorics

Cite

@article{arxiv.2412.16637,
  title  = {A lower bound on the Ramsey number $R_k(k+1,k+1)$},
  author = {Pavel Pudlák and Vojtěch Rödl and William J. Wesley},
  journal= {arXiv preprint arXiv:2412.16637},
  year   = {2026}
}

Comments

In this version we have replaced the 2-coloring of the bridge hypergraph found by a SAT-solver with a coloring defined by two equation in Z_3 found by Jack Wesley, who is now a coauthor

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

Abstract

Keywords

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