On the size of finite Sidon sequences

Abstract:

Let $h \geq 2$ be an integer. A set of positive integers B is called a ${B_h}$-sequence, or a Sidon sequence of order h, if all sums ${a_1} + {a_2} + \cdots + {a_h}$, where ${a_i} \in B (i = 1,2, \ldots ,h)$, are distinct up to rearrangements of the summands. Let ${F_h}(n)$ be the size of the maximum ${B_h}$-sequence contained in $\{ 1,2, \ldots ,n\}$. We prove that \[ {F_{2r - 1}}(n) \leq {({(r!)^2}n)^{1/(2r - 1)}} + O({n^{1/(4r - 2)}}).\]