Strictly monotonic multidimensional sequences and stable sets in pillage games

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.