Ramsey Numbers $R(K_3, G)$ for Graphs of Order 10

Abstract

In this article we give the generalized triangle Ramsey numbers $R(K_3, G)$ of 12 005 158 of the 12 005 168 graphs of order 10. There are 10 graphs remaining for which we could not determine the Ramsey number. Most likely these graphs need approaches focusing on each individual graph in order to determine their triangle Ramsey number. The results were obtained by combining new computational and theoretical results.  We also describe an optimized algorithm for the generation of all  maximal triangle-free graphs and triangle Ramsey graphs.  All Ramsey numbers up to 30 were computed by our implementation of this algorithm. We also prove some theoretical results that are applied to determine several triangle Ramsey numbers larger than 30. As not only the number of graphs is increasing very fast, but also the difficulty to determine Ramsey numbers, we consider it very likely that the table of all triangle Ramsey numbers for graphs of order 10 is the last complete table that can possibly be determined for a very long time.