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Section 11.2 Small 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.

\(n\) 3 4 5 6 7 8 9
\(m\)
3 6 9 14 18 23 36 39
4 18 25 36, 41 49, 61 58, 84 73, 115
5 43, 49 58, 87 80, 143 101, 216 126, 316
6 102, 165 113, 298 127, 495 169, 780
7 205, 540 217, 1031 241, 1713
8 282, 1870 317, 3583
9 565, 6588
Figure 11.3. Small Ramsey numbers \(R(m,n)\)

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.)