A note on the Erd\H{o}s-Szekeres theorem in two dimensions

TL;DR

This paper explores multidimensional extensions of the Erd ext{"o}s-Szekeres theorem, determining maximal array sizes avoiding certain monotone subarrays, and connects these findings to coding theory.

Contribution

It introduces new bounds for two-dimensional arrays avoiding specific monotone subarrays and links these results to a well-known problem in coding theory.

Findings

Maximal array sizes without certain monotone subarrays are determined.

Established equality of array sizes for different monotonicity conditions.

Connected array combinatorics with coding theory problems.

Abstract

Burkill and Mirsky, and Kalmanson, prove independently that, for every $r\ge 2, n\ge 1$, there is a sequence of $r^{2^n}$ vectors in $\mathbb R^n$, which does not contain a subsequence of $r+1$ vectors $v^1, v^2,\dots,v^{r+1}$ such that, for every $i$ between 1 and $n$, $(v^{j}_i)_{1\le j\le r+1}$ forms a monotone sequence. Moreover, $r^{2^n}$ is the largest integer with this property. In this short note, for two vectors $u = (u_1, u_2,\dots, u_n)$ and $v = (v_1, v_2, \dots, v_n)$ in $\mathbb R^n$, we say that $u\le v$ if, for every $i$ between 1 and $n$, $u_i\le v_i$. Just like Burkill and Mirsky, and Kalmanson, for every $k, \ell\ge 1, d\ge 2$ we find the maximal $N_1, N_2$ (which turn out to be equal) such that there are numerical two-dimensional arrays of size $(k+\ell-1)\times N_1$ and $(k+\ell)\times N_2$, which neither contain a subarray of size $k\times d$, whose columns form a non-decreasing sequence of $d$ vectors in $\mathbb R^k$, nor contain a subarray of size $\ell\times d$, whose columns form a non-increasing sequence of $d$ vectors in $\mathbb R^{\ell}$. In a consequent discussion, we consider a generalisation of this setting and make a connection with a famous problem in coding theory.