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arXiv · 1307.0843

New bounds for the distance Ramsey number

Andrey Kupavskii·Andrei Raigorodskii·Maria Titova
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Abstract

In this paper we study the distance Ramsey number $R_{{\it D}}(s,t,d)$. The \textit{distance Ramsey number} $R_{{\it D}}(s,t,d) $ is the minimum number $n$ such that for any graph $ G $ on $ n $ vertices, either $G$ contains an induced $ s $-vertex subgraph isomorphic to a distance graph in $ \Real^d $ or $ \bar {G} $ contains an induced $ t $-vertex subgraph isomorphic to the distance graph in $ \Real^d $. We obtain the upper and lower bounds on $R_{{\it D}}(s,s,d),$ which are similar to the bounds for the classical Ramsey number $R(\lceil \frac{s}{[d/2]} \rceil, \lceil \frac{s}{[d/2]} \rceil)$.

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Andrey Kupavskii, Andrei Raigorodskii, Maria Titova. 2013-07-02. New bounds for the distance Ramsey number. https://arxiv.org/abs/1307.0843

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