New bounds for the distance Ramsey number

TL;DR

This paper establishes new upper and lower bounds for the distance Ramsey number, a graph invariant related to induced subgraphs that are distance graphs in Euclidean space, extending classical Ramsey theory.

Contribution

The paper derives bounds for the distance Ramsey number $R_D(s,t,d)$, connecting it to classical Ramsey numbers and advancing understanding of geometric graph configurations.

Findings

Bounds on $R_D(s,s,d)$ are similar to classical Ramsey bounds.

Introduces new inequalities relating geometric and combinatorial Ramsey numbers.

Provides asymptotic estimates for large $s$ and fixed $d$.

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 $R(\lceil \frac{s}{[d/2]} \rceil, \lceil \frac{s}{[d/2]} \rceil)$.