New bounds for the distance Ramsey number

In this paper we study the distance Ramsey number R D ( s , t , d ) . The distance Ramsey number R 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 R d or G ̄ contains an induced t -vertex subgraph isomorphic to the distance graph in R d . We obtain the upper and lower bounds on R D ( s , s , d ) , which are similar to the bounds for the classical Ramsey number R ( ⌈ s [ d / 2 ] ⌉ , ⌈ s [ d / 2 ] ⌉ ) .