掠夺对策中的严格单调多维序列与稳定集
Strictly monotonic multidimensional sequences and stable sets in pillage games
摘要：
令$S \subset \mathbb{R}^n$的大小为$|S| > \ell^{2^n-1}$。我们表明，有不同的点$\{x^1,..., x^{\ell+1}\} \subset S$，使每个$i \in [n]$，坐标序列$(x^j_i)_{j=1}^{\ell+1}$是严格增加，严格减少，或常数，这对$|S|$的约束是最好的可能。这类似于$\real$中关于单调序列的\erdos -Szekeres定理。我们应用这些结果来界定一个稳定集的大小在抢劫游戏。我们还证明了一个定理的独立组合的利益。假设$\{a^1,b^1,...,a^t,b^t\}$是$\real^n$中的$2t$点的集合，使得不共享坐标的点对的集合精确地是$\{\{a^1,b^1\},...,\{a^t,b^t\}\}$。我们表明，$t \leq 2^{n-1}$，这个界限是最好的可能。
英文摘要：
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.
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