Classes of Graphs that Are Not Vertex Ramsey

Sauer [Combinatorics, 1 (1993), pp. 361--377] has conjectured that for any tree T and any clique K, the class Forb(T, K) of graphs that induces neither T nor K is not vertex Ramsey. This conjecture is implied by an even stronger conjecture of Gyarfas and independently by Sumner, that Forb(T, K) is $\chi$-bounded. Until now, for all trees T, if Forb(T, K) was known to not be vertex Ramsey, then Forb(T, K) was also known to be $\chi$-bounded. In this paper we introduce a new class of trees, spiders with toes, which includes all trees T such that Forb(T) is known to be $\chi$-bounded as well as other trees for which it is not known to be $\chi$-bounded. We show that for every spider with toes T, Forb(T, K) is not vertex Ramsey.