On the size of finite Sidon sequences

Let h > 2 be -an integer. A set of positive integers B is called a Bh-sequence, or a Sidon sequence of order h, if all sums aI + a2 + * + ah, where ai E B (i = 1, 2, ..., h), are distinct up to rearrangements of the summands. Let Fh(n) be the size of the maximum Bh-sequence contained in {1, 2, . n} . We prove that F2rI (n) 2 be an integer. A set of positive integers B is called a Bh-sequence if all sums a, + a2 + * *. + ah . where as E B (=1, 2,... ,h), are distinct up to rearrangements of the summands. A Bh-sequence is also called a Sidon sequence of order h [6]. We say that B is a Bh-sequence for Z/(n) if B is a finite Bh-sequence and all sums are distinct modulo n. Let Fh(n) denote the size of maximum Bh-sequences contained in the set of integers {1, 2, ... , n} and fh(n) the size of maximum Bh-sequence for Z/(n). Then it follows from a simple combinatorial argument that Fhf(n) 2, there exists a Bh-sequence B for Z/(mh 1) with IBI = m, where m is a prime power. This implies that Fhf(n) > (1 + o(1))nl/h. Therefore, F2(n) = (1. + o(l))vii. Erdos conjectured that F2 (n) = v/ii + 0(1) . For h = 3, Lee [4] obtained that 1/3 F3 (n) < I6 )4n) g2 n1 For h = 4, Lindstr6m [5] proved that F4(n) < (8n)1/4 + 0(n1/8). Received by the editors May 22, 1992 and, in revised form, September II, 1992. 1991 Mathematics Subject Classification. Primary 11B83; Secondary 1 1B50, 05B10.