Strictly monotonic multidimensional sequences and stable sets in pillage games

TL;DR

This paper extends the Erdős–Szekeres theorem to multidimensional sequences, establishing bounds on the size of sets with certain monotonic properties and applying these to stable sets in pillage games, along with a related combinatorial result.

Contribution

It introduces a multidimensional monotonic sequence theorem with optimal bounds and applies it to analyze stable sets in pillage games, also proving a new combinatorial bound on point pairings.

Findings

Existence of specific monotonic subsequences in large multidimensional point sets.

Bound on the size of stable sets in pillage games based on these monotonic sequence results.

A new combinatorial bound on point pairings with coordinate restrictions.

Abstract

Let $S \subset \mathbb{R}^n$ have size $|S| > \ell^{2^n-1}$. We show that there are distinct points $\{x^1,..., x^{\ell+1}\} \subset S$ such that for each $i \in [n]$, the coordinate sequence $(x^j_i)_{j=1}^{\ell+1}$ is strictly increasing, strictly decreasing, or constant, and that this bound on $|S|$ is best possible. This is analogous to the \erdos-Szekeres theorem on monotonic sequences in $\real$. We apply these results to bound the size of a stable set in a pillage game. We also prove a theorem of independent combinatorial interest. Suppose $\{a^1,b^1,...,a^t,b^t\}$ is a set of $2t$ points in $\real^n$ such that the set of pairs of points not sharing a coordinate is precisely $\{\{a^1,b^1\},...,\{a^t,b^t\}\}$. We show that $t \leq 2^{n-1}$, and that this bound is best possible.