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- W4300534353 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." @default.
- W4300534353 created "2022-10-03" @default.
- W4300534353 creator A5028282339 @default.
- W4300534353 date "2010-04-03" @default.
- W4300534353 modified "2023-09-27" @default.
- W4300534353 title "Strictly monotonic multidimensional sequences and stable sets in pillage games" @default.
- W4300534353 doi "https://doi.org/10.48550/arxiv.1004.0433" @default.
- W4300534353 hasPublicationYear "2010" @default.
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