On Finite Polarized Partition Relations

Abstract

Call an m × n array an m × n; k array if its mn entries come from a set of k elements. An m × n; 1 array has mn like entries. We write(1)if every m × n; k array contains a p × q; 1 sub-array. The negation of (1) is writtenand means that there is an m × n; k array containing no p × q; 1 sub-array. Relations (1) are called "polarized partition relations among cardinal numbers" by P. Erdös and R. Rado [2]. In this note we prove the following theorems.