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## Section11.2Small Ramsey Numbers

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.

For additional (or more current) data, see Dynamic Survey #DS1: “Small Ramsey Numbers” by Stanisław Radziszowski in the Electronic Journal of Combinatorics. (Figure 11.3 was last updated using the 12 January 2014 version of that article.)