Reprint from THE BULLETIN OF THE RESEARCH COUNCIL OF ISRAEL

A graph G is called complete if any two of its vertices are connected by an edge, a set of vertices of G are said to be independent if no two of them are connected by an edge . A well known theorem of Ramsay1 states that an infinite graph either contains an infinite complete subgraph or an infinite set of independent vertices . He also proved the finite form of this theorem, namely that there exists a function h(n) so that if G has h(n) or more vertices, then either G contains a complete graph of n vertices, or a set of n independent points . Another way of formulating the same theorem is to say that if the edges of a complete graph having h(n) vertices are coloured red or blue, then there exists either a red, or a blue complete graph of n vertices . Define f(k,l) as the least integer so that every graph of f(k,l) vertices either contains a complete graph of k vertices or contains a set of l independent points, f(k,l) has only been determined for small values of k and 1 2 . Szekeres3 proved that