. Every graph of chromatic number k with more than k(r \Gamma 1)(b \Gamma 1) vertices has a b-element independent set of vertices such that if any two of them are joined by an edge then the chromatic number stays the same or a r-element independent set of vertices such that joining any two of them by an edge increases the chromatic number. 0. Introduction. Let P be a class of graphs and let x and y be a pair of non-adjacent distinct vertices of G ffl P . For every graph consider a fixed coloring of all pairs fx; yg of nonadjacent vertices - we color e = fx; yg blue if G + feg belongs to P , and red - otherwise. A class, i.e., a property P of graphs will be called Ramseyan if large graphs from P contain large monochromatic subgraphs. It is easy to see that P is Ramseyan if and only if large enough graphs from P have large independent sets. The necessity of this condition is obvious, and the sufficiency follows from Ramsey the electronic journal of combinatorics 3 (1996), #R26 2 theo..