Geodesic
On 1-dependent ramsey numbers for graphs
Discussiones Mathematicae. Graph Theory, Tome 19 (1999) no. 1, pp. 93-110

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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.
Keywords: 1-dependence, irredundance, CO-irredundance, Ramsey numbers 
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Cockayne, E.; Mynhardt, C. On 1-dependent ramsey numbers for graphs. Discussiones Mathematicae. Graph Theory, Tome 19 (1999) no. 1, pp. 93-110. http://geodesic.mathdoc.fr/item/DMGT_1999_19_1_a7/