Ramsey Numbers

The Ramsey numbers are a sequence of numbers in combinatorics that answer the question of how large a group of people must be to guarantee that some fixed number of them are all mutual acquaintances or all mutual strangers, a formulation of the broader principle, named after the British mathematician and philosopher Frank Ramsey, that complete disorder is impossible and any sufficiently large structure must contain some highly organized substructure. The Ramsey number R(m,n) is the smallest number of vertices a complete graph must have so that, however its edges are colored in two colors, it is guaranteed to contain either a completely one colored subgraph of m vertices or a completely differently colored subgraph of n vertices. Even small Ramsey numbers are notoriously difficult to compute exactly; the value of R(5,5) remains unknown, known only to lie between 43 and 48, and the mathematician Paul Erdős famously remarked that humanity would be hard pressed to compute it even under the threat of alien invasion, given the astronomical size of the search involved.