On Erdős-Rado numbers

Abstract

In this paper new proofs of the Canonical Ramsey Theorem, which originally has been proved by Erdős and Rado, are given. These yield improvements over the known bounds for the arising Erdős-Rado numbersER(k; l), where the numbersER(k; l) are defined as the least positive integern such that for every partition of thek-element subsets of a totally orderedn-element setX into an arbitrary number of classes there exists anl-element subsetY ofX, such that the set ofk-element subsets ofY is partitioned canonically (in the sense of Erdős and Rado). In particular, it is shown that

$$2^{c1} .l^2 \leqslant ER(2;l) \leqslant 2^{c_2 .l^2 .\log l} $$

for every positive integerl≥3, wherec ₁,c ₂ are positive constants. Moreover, new bounds, lower and upper, for the numbersER(k; l) for arbitrary positive integersk, l are given.