GENERALIZED RAMSEY THEORY FOR GRAPHS - A SURVEY.

Abstract

This survey paper will emphasize the following class of problems: Given graphs G//1,. . . ,G//c, determine or estimate the Ramsey number r(G//1,. . . ,G//c), the smallest number p such that if the lines of a complete graph K//p are c-colored in any manner, then for some j there exists a subgraph in color j which is isomorphic to G//j. Ramsey numbers have now been evaluated completely in a large number of cases, particularly when c equals 2. The most strikingly general result is due to V. Chvatal: If T is a tree on m points, then r(Tk,K//n) equals mn-m-n plus 2. Also of interest is the study of asymptotic questions about r(G//1,. . . ,G//c).