The third author proved earlier [8] that if a Euclidean space is colored with red and blue so that the distance one is forbidden for blue, and translates of some \(k\)-point configuration are forbidden for red, then the unit-distance chromatic number of the space is no greater than \(k\). Here we give a generalization.
Citation
D.G. Hoffmant, P.D. Johnson Jr, A.D. Szlam. A New Lemma in Ramsey Theory[J], Journal of Combinatorial Mathematics and Combinatorial Computing, Volume 038. 123-128. .