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