On Finite Polarized Partition Relations

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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 written and 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.

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