Let
n
n
random points be chosen independently from the uniform distribution on the unit
k
k
-cube
[
0
,
1
]
k
{[0,1]^k}
. Order the points coordinate-wise and let
H
k
(
n
)
{{\mathbf {H}}_k}\left ( n \right )
be the cardinality of the largest chain in the resulting partially ordered set. We show that there are constants
c
1
,
c
2
,
…
{c_1},{c_2}, \ldots
such that
c
k
>
e
,
lim
k
→
∞
c
k
=
e
{c_k} > e,\;{\lim _{k \to \infty }}{c_k} = e
, and
lim
n
→
∞
H
k
(
n
)
/
n
1
/
k
=
c
k
{\lim _{n \to \infty }}{{\mathbf {H}}_k}\left ( n \right )/{n^{1/k}} = {c_k}
in probability. This generalizes results of Hammersley, Kingman and others on Ulam’s ascending subsequence problem, and settles a conjecture of Steele.