Directory for known bound of Ramsey numbers?

Question

I must admit I'm not a google connoisseur, but I have not been able to find a place where I can find known lower bounds for many Ramsey numbers, something ideal would be if I could insert (3,44) and receive the two best known bounds to the problem. Where can I find this? Does it exist?

Answer 1

The dynamic survey "Small Ramsey Numbers", maintained by Stanisław P. Radziszowski, is an essential reference.

You ask in the comments about $R(3,45)$. The paper contains a table of lower bounds for $R(3,k)$ with $k \leq 38$, suggesting $R(3,45)$ has not been investigated explicitly. You will have to instead look at more general results for $R(3,k)$. For example, I find on page 8 (item c) $$R(3,4k+1)≥6R(3,k+1)−5.$$

For your example, this means $$R(3,45)≥6R(3,12)−5≥6⋅52−5=307.$$

Page 9 (item 3) states $$ R(3,k) \leq \frac{k^2}{\log k}, $$ which gives $$ R(3,45) \leq 531. $$

Taken together, we have $$ 307 \leq R(3,45) \leq 531. $$