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?
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$\begingroup$ See here and here. $\endgroup$– Andrés E. CaicedoJun 9, 2014 at 5:36
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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 $$ 307 \leq R(3,45) \leq 531. $$
answered Jun 7, 2014 at 19:31
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$\begingroup$ Thanks, I read this, but isn't there a website which is just a big table, I want known bounds for $R(3,45)$ Where can I find this? $\endgroup$– AsinomásJun 7, 2014 at 19:43
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1$\begingroup$ @Bananarama given the lack of google results for such a table I guess it probably doesn't exist and your best bet is the paper mentioned. Item (3) on page 9 then seems to be the best known bound for $R(3,45)$ based on a quick look. $\endgroup$ Jun 8, 2014 at 13:47