On 1-dependent ramsey numbers for graphs

A set X of vertices of a graph G is said to be 1-dependent if the subgraph of G induced by X has maximum degree one. The 1-dependent Ramsey number t1(l, m) is the smallest integer n such that for any 2-edge colouring (R, B) of Kn, the spanning subgraph B of Kn has a 1-dependent set of size l or the subgraph R has a 1-dependent set of size m. The 2-edge colouring (R, B) is a t1(l, m) Ramsey colouring of Kn if B (R, respectively) does not contain a 1-dependent set of size l (m, respectively); in this case R is also called a (l, m, n) Ramsey graph. We show that t1(4, 5) = 9, t1(4, 6) = 11, t1 (4, 7) = 16 and t1(4, 8) = 17. We also determine all (4,4,5), (4,5,8), (4,6,10) and (4,7,15) Ramsey graphs.