ON ERDOS-RADO NUMBERS

In this paper new proofs of the Canonical Ramsey Theorem, which originally has been proved by Erdos and Rado, are given. These yield improvements over the known bounds for the arising Erdos-Rado numbers ER(k;l), where the numbers ER(k;l) are defined as the least positive integer n such that for every partition of the Ic-element subsets of a totally ordered n-element set X into an arbitrary number of classes there exists an I-element subset Y of X, such that the set of k-element subsets of Y is partitioned canonically (in the sense of Erdos and Rado). In particular, it is shown that 2(c1 . l2) less than or equal to ER(2;1) less than or equal to 2(c2 . l2). log l for every positive integer l greater than or equal to 3, where c1,c2 are positive constants. Moreover, new bounds, lower and upper, for the numbers ER(k;l) for arbitrary positive integers k, l are given.