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arXiv · 1910.13318

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

M. Bucić·B. Sudakov·T. Tran
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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.

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M. Bucić, B. Sudakov, T. Tran. 2019-10-29. Erd\H{o}s-Szekeres theorem for multidimensional arrays. https://arxiv.org/abs/1910.13318

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