How do mathematics go about finding larger Ramsey numbers such as R(5, 5)? How do they find upper bounds on these numbers?
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Some ideas:
A famous upper bound for any $k,l$ is from Erdos-Szekeres:
$R(k,l) \le \binom {k+l-2} {k-1}$.
In your case, $k=l=5$, so $R(5,5) \le \binom {8}{4}$
You can find some thoughts of Paul Erdos about Ramsey numbers in that sheet aswell. It is generally hard to find such big numbers in a very exact way, we don't have many methods to do that.
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$\begingroup$ The Erdos-Szekeres conjecture doesn't seem to be proven. Has it been? $\endgroup$ – Jimmy360 May 23 '15 at 21:04
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$\begingroup$ en.wikibooks.org/wiki/Combinatorics/Bounds_for_Ramsey_numbers Here you can find it. $\endgroup$ – Atvin May 23 '15 at 21:05
