  The classical Erd\H{o}s-Szekeres theorem dating back almost a hundred years
states that any sequence of $(n-1)^2+1$ distinct real numbers contains a
monotone subsequence of length $n$. This theorem has been generalised to higher
dimensions in a variety of ways but perhaps the most natural one was proposed
by Fishburn and Graham more than 25 years ago. They defined the concept of a
monotone and a lex-monotone array and asked how large an array one needs in
order to be able to find a monotone or a lex-monotone subarray of size $n
\times \ldots \times n$. Fishburn and Graham obtained Ackerman-type bounds in
both cases. We significantly improve these results. Regardless of the dimension
we obtain at most a triple exponential bound in $n$ in the monotone case and a
quadruple exponential one in the lex-monotone case.
