On Pattern Ramsey Numbers of Graphs

A color pattern is a graph whose edges are partitioned into color classes. A family F of color patterns is a Ramsey family if there is some integer N such that every edgecoloring of KN has a copy of some pattern in F . The smallest such N is the (pattern) Ramsey number R(F ) of F . The classical Canonical Ramsey Theorem of Erdős and Rado [4] yields an easy characterization of the Ramsey families of color patterns. In this paper we determine R(F ) for all families consisting of equipartitioned stars, and we prove that 5b s−1 2 c+1 ≤ R(F ) ≤ 3s− √ 3s when F consists of a monochromatic star of size s and a polychromatic triangle.