If there are R(m, n) people in a room then either m of them are entirely mutually acquainted among each other or n of them of whom none knows any of the other n-1.

the minimum number of points a graph must have that forces it to have a subgraph isomorphic to K_m or to have n isolated points

If every line in a complete graph with this many points are coloured with one of two colours, then there will be always be a complete subgraph of order m with all lines of one colour or a complete graph of order n with all lines of the other colour

R is a bipartite graph

It is the Ramsay number - the minimum number of vertices required to force all such graphs to contain K_m or