Given the integers l 1, k 1, l 2, k 2, r, which satisfy the condition l 1, l 2 ⩾ r ⩾ k 1, k 2 > 0, we define m = N ( l 1, k 1; l 2, k 2; r) as the smallest integer with the following property: if S is a set containing m points and the r-subsets of S are partitioned arbitrarily into two classes, then for i = 1 or 2 there exists an l i subset of S each of whose k i -subsets lies in some r-subset of the i th class. The integers defined in this way form a collection of which the usual Ramsey numbers are a special case: i.e., the Ramsey number N( l 1, l 2; r) is represented as N( l 1, r; l 2, r; r). We derive two major results concerning the values of these generalized Ramsey numbers. If k 1 + k 2 = r + 1 then N( l 1, k 1; l 2, k 2; r) = l 1 + l 2 − k 1 − k 2 + 1, corresponding to the “pigeonhole principle”. For k 1 + k 2 ⩽ r, we show that N( l 1, k 1; l 2, k 2; r) = max ( l 1, l 2). The next interesting case occurs for k 1 + k 2 = r + 2, where we show that there are constants c 1 and c 2 such that for sufficiently large l, 2 c 1 l < N( l, k 1; l, k 1; r) < 2 c 2 l .