Erdős-Szekeres theorem for multidimensional arrays

The classical Erdős-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.