Papers › Tighter Bounds on Directed Ramsey Number R(7)
Tighter Bounds on Directed Ramsey Number R(7)
David Neiman, John Mackey, Marijn Heule
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Tournaments are orientations of the complete graph, and the directed Ramsey number R(k) is the minimum number of vertices a tournament must have to be guaranteed to contain a transitive subtournament of size k, which we denote by TTₖ. We include a computer-assisted proof of a conjecture by Sanchez-Flores that all TT₆-free tournaments on 24 and 25 vertices are subtournaments of ST₂₇, the unique largest TT_6-free tournament. We also classify all TT₆-free tournaments on 23 vertices. We use these results, combined with assistance from SAT technology, to obtain the following improved bounds: 34 ≤R(7) ≤47.
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