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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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