Vol. LXXIV, 2 (2005)
p. 243 - 254

A Number-Theoretic Conjecture and its Implication for Set Theory
L. Halbeisen

Abstract. For any set S let |seq 1-1(S)| denote the cardinality of the set of all finite one-to-one sequences that can be formed from S, and for positive integers a let |aS| denote the cardinality of all functions from S to a. Using a result from combinatorial number theory, Halbeisen and Shelah have shown that even in the absence of the axiom of choice, for infinite sets S one always has |seq 1-1 (S)| ¹ |2S| (but nothing more can be proved without the aid of the axiom of choice). Combining stronger number-theoretic results with the combinatorial proof for a = 2, it will be shown that for most positive integers a one can prove the inequality |seq 1-1 (S)| ¹ |aS| without using any form of the axiom of choice. Moreover, it is shown that a very probable number-theoretic conjecture implies that this inequality holds for every positive integer a in any model of set theory.

Keywords:   Non-repetitive sequences, axiom of choice, combinatorial number theory.

AMS Subject classification:  Primary: 11B50; Secondary: 03E05, 11B75, 11K31.

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