In [2, 3], Chung and Liu introduce the following generalization of Ramsey Theory for graphs. Choose c colors, and integers d 1, d 2,…, d n satisfying 1⩽ d 1< d 2<⋯< d n < c. Order the ( c d 1) subsets of d 1 colors, ( c d 2) subsets of d 2 colors,…,( c d 2) subsets of d n colors and let t=∑ i ( c d i . For graphs G 1, G 2,…, G 3, the ( d 1, d 2,…, d n )-chromatic Ramsey number denoted by R c d 1, d 2, …, d n ( G 1, G 2, …, G t ), is the smallest integer p such that if the edges of K p are colored with c colors in any fashion, then for some i the subgraph whose edges are colored with the ith subset of colors contains G i . The numbers R 2 1( G 1, G 2), simply denoted R( G 1, G 2), have been surveyed in [1] and in particular if G 1, G 2,…, G c are complete graphs, then R c 1( G 1, G 2,…, G c ), denoted R( G 1, G 2,…, G c ), are the classical Ramsey numbers [7]. Chung and Liu have determined some numbers of the form R 3 2( K i , K j , K m ), R 3 1.2( K i 1, K i 2,…, K i 6) and R 4 1,3( K i 1,…, K i 8). In this paper we are mainly concerned with the numbers R 4 1.2( K i 1, K i 2,…, K i 10).