R(5, 22) and R(6, 21)

A quick post about small Ramsey numbers. I like to write on my blog about things which I do not intend to publish, but also do not want to keep as private knowledge. This is one of these posts.

Stanisław P. Radziszowski writes the following in the 15th revision of his survey on small Ramsey numbers:

One can expect that the lower bounds in Table II are weaker than those in Table I, especially smaller ones, in the sense that some of them should not be that hard to improve, in contrast to the bounds in Table I.

Table II contains slightly larger Ramsey numbers, for instance R(5, 22) and R(6, 21). So when one has idle CPU time, it seems to be reasonable to use it here, I thought some weeks ago. And indeed, it is. I found a witness for R(5, 22) \geq 492 and a witness for for R(6, 21) \geq 884. This improves the previously known lower bounds by a gigantic 7 in the first case and a nearly as gigantic 6 in the second case. Here you can find a short description of both cases.

And if anyone asks: The upper bounds are much, much, much larger than both lower bounds. Also I did not put any effort into my search, so it is probably very feasible to improve a few more numbers in Table II of the survey.

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