Canonical Pattern Ramsey Numbers

Abstract

A color pattern is a graph whose edges have been partitioned into color classes. A family [InlineMediaObject not available: see fulltext.] of color patterns is a Ramsey family provided there is some sufficiently large integer N such that in any edge coloring of the complete graph K N there is an (isomorphic) copy of at least one of the patterns from [InlineMediaObject not available: see fulltext.]. The smallest such N is the Ramsey number of the family [InlineMediaObject not available: see fulltext.]. The classical Canonical Ramsey theorem of Erd?s and Rado asserts that the family of color patterns is a Ramsey family if it consists of monochromatic, rainbow (totally multicolored) and lexically colored complete graphs. In this paper we treat the asymmetric case by studying the Ramsey number of families containing a rainbow triangle, a lexically colored complete graph and a fixed arbitrary monochromatic graph. In particular we give asymptotically tight bounds for the Ramsey number of a family consisting of rainbow and monochromatic triangle and a lexically colored K N . Among others, we prove some canonical Ramsey results for cycles.