TITLE: Finding higher Ramsey numbers
QUESTION [5 upvotes]: How do mathematics go about finding larger Ramsey numbers such as R(5, 5)? How do they find upper bounds on these numbers?

REPLY [2 votes]: 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.