Erd\H{o}s-Szekeres theorem for multidimensional arrays

TL;DR

This paper extends the Erdős-Szekeres theorem to multidimensional arrays, providing significantly improved bounds on the size needed to guarantee monotone or lex-monotone subarrays in higher dimensions.

Contribution

The authors significantly improve existing Ackerman-type bounds for the size of arrays needed to find monotone or lex-monotone subarrays in multiple dimensions.

Findings

At most triple exponential bound for monotone subarrays

At most quadruple exponential bound for lex-monotone subarrays

Bounds are dimension-independent and significantly tighter than previous results

Abstract

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.