Table of Contents
International Journal of Combinatorics
Volume 2014 (2014), Article ID 826141, 11 pages
http://dx.doi.org/10.1155/2014/826141
Research Article

On the General Erdős-Turán Conjecture

1Université Jean Monnet, 23 rue Michelon, 42023 Saint-Étienne, France
2120 rue de Charonne, 75011 Paris, France
3Penn State University, 25 Yearsley Mill Road, Media, PA 19063, USA
4Kyläkunnantie 53, 00660 Helsinki, Finland

Received 28 July 2014; Accepted 14 October 2014; Published 17 November 2014

Academic Editor: Chris A. Rodger

Copyright © 2014 Georges Grekos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The general Erdős-Turán conjecture states that if is an infinite, strictly increasing sequence of natural numbers whose general term satisfies , for some constant and for all , then the number of representations functions of is unbounded. Here, we introduce the function , giving the minimum of the maximal number of representations of a finite sequence of natural numbers satisfying for all . We show that is an increasing function of and that the general Erdős-Turán conjecture is equivalent to . We also compute some values of . We further introduce and study the notion of capacity, which is related to the function by the fact that is the capacity of the set of squares of positive integers, but which is also of intrinsic interest.