发表机构
Carnegie Mellon University; University of Rochester（卡内基梅隆大学; 罗切斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过拓扑Figiel-Lindenstrauss-Milman不等式证明实射影空间三角剖分顶点数的超多项式下界，并应用于图论和符号复形指标估计。
AI 中文摘要
我们证明实射影$d$-空间的每个单纯三角剖分都有$\exp(\Omega(\sqrt d))$个顶点。结合已知构造，这确定了最小顶点数为$\mu_d=\exp(d^{1/2+o(1)})$。该结果源于Figiel--Lindenstrauss--Milman不等式的一个拓扑推广，回答了Frick、Hosseini和Vasileuski最近提出的一个问题：一个具有自由胞腔对合、$v$个顶点和$f$个极大胞腔的有限强正则CW复形，其$\mathbb{Z}/2$-指标至多为$O(\log v\log f)$。我们通过约束Hessian的迹估计来界定Morse胞腔的维数，从而为中心对称多胞形的经典不等式提供了一个Morse理论证明。作为该不等式的进一步应用，我们给出了无三角形拓扑$t$-色图的阶的$\exp(\Omega(\sqrt t))$下界，并将符号复形的指标界定为：对于全矩阵为$O(d\log^2 N)$，对于部分矩阵为$O(d\log^3 N)$，其中$N\geq2$是列数，$d\geq1$是VC维。
英文摘要
We prove that every simplicial triangulation of real projective $d$-space has $\exp(Ω(\sqrt d))$ vertices. Together with known constructions, this determines the minimum vertex number as $μ_d=\exp(d^{1/2+o(1)})$. The result follows from a topological generalization of the Figiel--Lindenstrauss--Milman inequality, answering a recent question of Frick, Hosseini, and Vasileuski: a finite strongly regular CW complex with a free cellular involution, $v$ vertices, and $f$ maximal cells has $\mathbb{Z}/2$-index at most $O(\log v\log f)$. We bound the dimensions of Morse cells by a trace estimate for a constrained Hessian, obtaining a Morse-theoretic proof of the classical inequality for centrally symmetric polytopes. As further applications of this inequality, we give an $\exp(Ω(\sqrt t))$ lower bound for the order of a triangle-free topologically $t$-chromatic graph and bound the index of sign complexes by $O(d\log^2 N)$ for total matrices and $O(d\log^3 N)$ for partial matrices, where $N\geq2$ is the number of columns and $d\geq1$ is the VC dimension.
Comments16 pages
