关于高维多面体不可避免的面
On Unavoidable Faces of High-Dimensional Polytopes
AI总结：
本文证明简单多面体上的阈值f_s(ℓ,k)在ℓ≥2且k≥3时有限，改进了3-面规模的下界，其证明需用线性规划计算标记数的有理证书。
AI中文摘要：
Kalai的立方体-单形猜想断言：对所有正整数ℓ,k，存在整数f(ℓ,k)，使得每个维数至少为f(ℓ,k)的多面体要么有一个单形ℓ-面，要么有一个立方体k-面；记f_s(ℓ,k)为限制在简单多面体上的阈值。目前仅当ℓ,k≤2时已知f(ℓ,k)是有限的，此外Kalai证明f_s(2,k)≤2k²。本文证明：对所有ℓ≥2且k≥3，f_s(ℓ,k)是有限的，这是ℓ>2时的首个此类结果，其中f_s(2,k)≤2k²-1，且当ℓ≥3时f_s(ℓ,k)≤(1/2)k²ℓ2^k。在反向方向，本文得到下界f(ℓ,k)≥(5⌊ℓ/2⌋ + (ℓ mod 2) - 1)(k-1)+1，以及f_s(ℓ,k)≥max{4,2(ℓ-1)}(k-1)+1。一个相关问题是求高维多面体中3-面的最小可能规模，Meisinger、Kleinschmidt和Kalai证明每个维数d≥9的有理d-多面体都有一个顶点数少于78或面数少于78的3-面，本文改进了该界：每个维数至少为15的凸多面体都有一个面数至多为13的3-面。本文证明的一个步骤需要关于标记数的显式精确有理证书或恒等式，该证书通过线性规划计算得到。
英文摘要：
Kalai's cube--simplex conjecture asserts that for all positive integers $\ell,k$, there is an integer $f(\ell,k)$ such that every polytope of dimension at least $f(\ell,k)$ has either a simplex $\ell$-face or a cube $k$-face; let $f_s(\ell,k)$ denote the threshold restricted to simple polytopes. Finiteness of $f(\ell,k)$ is known only for $\ell,k \leq 2$. In addition, Kalai proved that $f_s(2,k) \leq 2k^2$. Here we prove that $f_s(\ell,k)$ is finite for all $\ell \geq 2$ and $k \geq 3$, the first such result beyond $\ell = 2$, with $f_s(2,k) \leq 2k^2-1$ and $f_s(\ell,k) \leq \tfrac{1}{2}k^2\ell\,2^k$ for $\ell \geq 3$. In the opposite direction, we obtain the lower bounds $f(\ell,k) \geq (5\lfloor \ell/2 \rfloor + (\ell \bmod 2) - 1)(k-1)+1$ and $f_s(\ell,k) \geq \max\{4,\,2(\ell-1)\}(k-1)+1$. A companion question asks for the minimum possible size of a 3-face within a higher-dimensional polytope. Meisinger, Kleinschmidt and Kalai proved that every rational $d$-polytope with $d \geq 9$ has a $3$-face with fewer than $78$ vertices or fewer than $78$ facets. Here we improve their bound: every convex polytope of dimension at least $15$ has a $3$-face with at most $13$ facets. One step of our proof requires an explicit exact rational certificate or identity on flag numbers. This certificate is computed using linear programming.
