Generalized Ramsey numbers for paths in 2 -chromatic graphs

Chung and Liu have defined the d -chromatic Ramsey number as follows. Let 1 ≤ d ≤ c and let t = ( c d ) . Let 1 , 2 , … , t be the ordered subsets of d colors chosen from c distinct colors. Let G 1 , G 2 , … , G t be graphs. The d -chromatic Ramsey number denoted by r d c ( G 1 , G 2 , … , G t ) is defined as the least number p such that, if the edges of the complete graph K p are colored in any fashion with c colors, then for some i , the subgraph whose edges are colored in the i th subset of colors contains a G i . In this paper it is shown that r 2 3 ( P i , P j , P k ) = [ ( 4 k + 2 j + i − 2 ) / 6 ] where i ≤ j ≤ k < r ( P i , P j ) , r 2 3 stands for a generalized Ramsey number on a 2 -colored graph and P i is a path of order i .