We show that, in any colouring of the edges of K_53 with two colours, there exists a monochromatic K_5, and hence R(5,5) <= 53. This is accomplished in three stages: a full enumeration of edges in (4,5)-good graphs, and a proof of the nonexistence of (5,5)-good graphs on 53 vertices. Only the first stage required extensive help from the computer.
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Australasian Journal of Combinatorics 5 (1992) 13-20
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