Further improving of upper bound on a geometric Ramsey problem

Eryk Lipka

Further improving of upper bound on a geometric Ramsey problem

Combinatorics 2020-04-14 v2

Authors: Eryk Lipka

Abstract

We consider following geometric Ramsey problem: find the least dimension $n$ such that for any 2-coloring of edges of complete graph on the points $\{\pm 1\}^n$ there exists 4-vertex coplanar monochromatic clique. Problem was first analyzed by Graham and Rothschild and they gave an upper bound: $n\le F(F(F(F(F(F(F(12)))))))$ , where $F(m) = 2\uparrow^m3$ . In 2014 Lavrov, Lee and Mackey greatly improved this result by giving upper bound $n< 2\uparrow\uparrow\uparrow 6 < F(5)$ . In this paper we revisit their estimates and reduce upper bound to $n< 2\uparrow\uparrow\uparrow 5$

Keywords

ramsey numbers ramsey theory rainbow coloring

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@article{arxiv.1905.05617,
  title  = {Further improving of upper bound on a geometric Ramsey problem},
  author = {Eryk Lipka},
  journal= {arXiv preprint arXiv:1905.05617},
  year   = {2020}
}

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3 pages

Further improving of upper bound on a geometric Ramsey problem

Abstract

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