AI 中文总结
研究VC维的多彩变体，通过该概念得到多种几何结果，包括改进的特弗伯格定理、多彩\(k\)重特弗伯格定理等，还扩展到凸集并集的多彩特弗伯格定理，提升了相关定量界。
AI 中文摘要
VC维是集合系统复杂性的基本度量。本文引入并研究了VC维的多彩变体，用于刻画有色基集上集合系统的行为。通过研究这一新概念，得到多种几何结果。首先证明了具有拉东数\(D\)的可分抽象凸空间存在特弗伯格数为\(O(D^2r\log r)\)的特弗伯格定理，改进了已有界。其次证明了可分抽象凸空间的首个多彩\(k\)重特弗伯格定理，并由此得到多个相关定理，其定量界优于之前已知的。最后将方法扩展到凸集并集的多彩特弗伯格定理。
英文摘要
The VC-dimension is a fundamental measure of the complexity of a set system. In this paper, we introduce and study a colorful variant of VC-dimension that captures the behavior of set systems on colored ground sets. By studying this new notion, we obtain a variety of geometric results. First, we prove that separable abstract convexity spaces with Radon number $D$ admit a Tverberg theorem with Tverberg number $O(D^2 r \log r)$. This bound significantly improves the $O(Dr^2\log r)$ bound of Alon and Smorodinsky from SODA'26 and is the first quasi-linear bound in $r$, in which the dependence on $D$ is not super-exponential. Second, we prove the first colorful $k$-wise Tverberg theorem for separable abstract convexity spaces. Using this theorem, we obtain a colorful selection lemma with $O(D^3)$ colors, an uncolored selection lemma for subsets of size $O(D^3)$, a weak $\varepsilon$-net theorem with nets of size $O_D(\varepsilon^{-O(D^3)})$, and a $(p,q)$-theorem with exponent of $\mathrm{poly}(D)$. All these quantitative bounds are significantly better than the best previously known general bounds for abstract convexity spaces. Finally, we extend our method to obtain a colorful Tverberg theorem for unions of convex sets, generalizing the uncolored theorem of Alon and Smorodinsky (SODA'26).
Comments22 pages
