# Finding higher Ramsey numbers

How do mathematics go about finding larger Ramsey numbers such as R(5, 5)? How do they find upper bounds on these numbers?

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 $R(5,5) \le \binom {8}{4}$