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Define  to be the least integer  such that, in any red-blue 2-coloring of the edges of a  -vertex complete graph  , there must exist either an all-red  or an all-blue  .
Frank Ramsey proved these numbers always exist. He famously pointed out that among any 6 people, some three are mutual friends or some three mutual non-friends. That is,
 . Since a red pentagon with a blue pentagram drawn inside it has no monochromatic triangle,
 . So  .
Special attention is usually paid to the diagonal  , which is often just written  .
One can also generalize this in various ways, e.g. consider  for three-colorings of edges, etc (any number of arguments), and allow general sets of graphs, not just pairs of complete ones.
Ramsey numbers are very difficult to determine. To prove lower bounds, construct good edge-colorings of some  and, use a clique-finder to find the largest mono-colored cliques. To prove upper bounds, the main tool has been
 which implies
 and then
![$ R(k) \le [1+o(1)] 4^k / (4 \sqrt{\pi k}$](http://images.planetmath.org:8080/cache/objects/8387/l2h/img16.png "$ R(k) \le [1+o(1)] 4^k / (4 \sqrt{\pi k}$") when
 . From considering random colorings and using a probabilistic nonconstructive existence argument, one may show
![$ R(k) \ge k 2^{k/2} [ o(1) + \sqrt{2} / e ]$](http://images.planetmath.org:8080/cache/objects/8387/l2h/img18.png "$ R(k) \ge k 2^{k/2} [ o(1) + \sqrt{2} / e ]$") . It is known that  ,  ,  ,  , and
 . For a survey of the best upper and lower bounds available on small Ramsey numbers, see Radziszowski's survey (alternate link). Another kind of Ramsey-like number which has not gotten as much attention as it deserves, are Ramsey numbers for directed graphs. Let
 denote the least integer  so that any tournament (complete directed graph with singly-directed arcs) with  vertices contains an acyclic (also called “transitive”)
 -node tournament. (Analogies: 2-color the edges  two directions for arcs. Monochromatic  acyclic, i.e. all arcs “point one way.”)
Again, to prove lower bounds, construct good tournaments and apply something like a clique-finder (but instead aimed at trying to find the largest acyclic induced subgraph). To prove upper bounds, the main tool is
 . That can be used to show the upper bound, and random-orientation arguments combined with a nonconstructive probabilistic existence argument show the lower bound, in
![$ [1+o(1)] 2^{n+1)/2} \le \vec{R}(n) \le 55 \cdot 2^{n-7}$](http://images.planetmath.org:8080/cache/objects/8387/l2h/img31.png "$ [1+o(1)] 2^{n+1)/2} \le \vec{R}(n) \le 55 \cdot 2^{n-7}$") . It is known that
 ,
 ,
 ,
 ,
 ,
 , and
 . For a full survey of directed graph Ramsey numbers includng proofs and refererences, see Smith's survey.
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