The analogue of Erdős–Turán conjecture in

Abstract: Given a set let denote the number of ordered pairs such that . The celebrated Erdős–Turán conjecture states that if such that for all sufficiently large n, then the representation function must be unbounded. For each positive integer m, let be the least positive integer r such that there exists a set with and . Ruzsa''s method in [I.Z. Ruzsa, A just basis, Monatsh. Math. 109 (1990) 145–151] implies that must be bounded. It is pleasure to call a Ruzsa''s number. In this paper we prove that all Ruzsa''s numbers . This improves the previous bound . Several related open problems are proposed. Video abstract: For a video summary of this paper, please visit http://www.youtube.com/watch?v=hgDwkwg_LzY. [Copyright &y& Elsevier]