Actually determining the Ramsey numbers $$R(m,n)$$ referenced in Theorem 11.2 seems to be a notoriously difficult problem, and only a handful of these values are known precisely. In particular, $$R(3,3)=6$$ and $$R(4,4)=18\text{,}$$ while $$43\le R(5,5)\le 49\text{.}$$ The distinguished Hungarian mathematician Paul Erdős said on many occasions that it might be possible to determine $$R(5,5)$$ exactly, if all the world's mathematical talent were to be focused on the problem. But he also said that finding the exact value of $$R(6,6)$$ might be beyond our collective abilities.
In the following table, we provide information about the Ramsey numbers $$R(m,n)$$ when $$m$$ and $$n$$ are at least $$3$$ and at most $$9\text{.}$$ When a cell contains a single number, that is the precise answer. When there are two numbers, they represent lower and upper bounds.
www.combinatorics.org/ojs/index.php/eljc/article/view/DS1