Generalized Ramsey numbers for paths in 2-chromatic graphs

Abstract

Chung and Liu have defined the d-chromatic Ramsey number as follows. Let 1≤d≤c and let t=(cd). Let 1,2,…,t be the ordered subsets of d colors chosen from c distinct colors. Let G1,G2,…,Gt be graphs. The d-chromatic Ramsey number denoted by rdc(G1,G2,…,Gt) is defined as the least number p such that, if the edges of the complete graph Kp are colored in any fashion with c colors, then for some i, the subgraph whose edges are colored in the ith subset of colors contains a Gi. In this paper it is shown that r23(Pi,Pj,Pk)=[(4k+2j+i−2)/6] where i≤j≤k<r(Pi,Pj), r23 stands for a generalized Ramsey number on a 2-colored graph and Pi is a path of order i.

Keywords

• Ramsey number
• generalized Ramsey number
• d-chromatic Ramsey number
• colored graph.