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 t 1 (l, m) is the smallest integer n such that for any 2-edge colouring (R, B) of K n , the spanning subgraph B of K n 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 t 1 (l, m) Ramsey colouring of K n if B (R, respectively) does not contain a 1-dependent set of size l (m,

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... y); in this case R is also called a (l, m, n) Ramsey graph. We show that t 1 (4, 5) = 9, t 1 (4, 6) = 11, t 1 (4, 7) = 16 and t 1 (4, 8) = 17. We also determine all (4,4,5), (4,5,8), (4,6,10) and (4,7,15) Ramsey graphs.