Here - Combinatorial algorithms and algorithmic graph theory

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Summary
Afterwards we show how this algorithm can be extended to enumerate triangle
Ramsey graphs efficiently. This algorithm allowed us to determine all triangle
Ramsey numbers up to 30 for graphs of order 10. By combining our computational
results with new theoretical results, we were able to determine the triangle
Ramsey number of nearly all of the 12 005 168 graphs of order 10, except for 10
of the hardest cases. Because of the rapid growth of Ramsey numbers, the list of
triangle Ramsey numbers for graphs of order 10 will very likely be the last list of
Ramsey numbers that can be completed for a very long time.
In the second part of this chapter we develop completely different specialised
algorithms to improve the upper bounds of classical triangle Ramsey numbers.
These are triangle Ramsey numbers R(K 3 , G) where the graph G is a complete
graph. Using these algorithms we managed to determine improved upper bounds
for several classical triangle Ramsey numbers. More specifically, we proved
that R(K 3 , K 10 ) ≤ 42, R(K 3 , K 11 ) ≤ 50, R(K 3 , K 13 ) ≤ 68, R(K 3 , K 14 ) ≤ 77,
R(K 3 , K 15 ) ≤ 87 and R(K 3 , K 16 ) ≤ 98. All of these new upper bounds improve
the old upper bounds by one. We also determine all critical Ramsey graphs for
K 8 and prove that the known Ramsey graph for K 9 is unique. A Ramsey graph
for G is critical if it has R(K 3 , G) − 1 vertices.