A Colorful Extension of VC-dimension and Geometric Applications

2026-07-11

AI Summary

This work extends VC dimension theory to colored set systems by introducing the novel notion of “colored VC dimension” to capture the combinatorial complexity of set systems over colored ground sets. Building on this framework and integrating tools from abstract convexity, Radon numbers, and Tverberg-type theorems, the paper establishes the first combinatorial geometric analysis method tailored to colored structures. Key contributions include the first k-wise colored Tverberg theorem in separable abstract convexity spaces, a Tverberg number bound of \(O(D^2 r \log r)\), an \(O(D^3)\)-color selection lemma, a weak \(\varepsilon\)-net of size \(O_D(\varepsilon^{-O(D^3)})\), and a \((p,q)\)-theorem with exponent \(\mathrm{poly}(D)\). All quantitative bounds substantially improve upon existing general-purpose results.

Abstract

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).

Problem

• VC-dimension
• colorful
• Tverberg theorem
• abstract convexity spaces
• geometric combinatorics

Innovation

• colorful VC-dimension
• Tverberg theorem
• abstract convexity spaces
• selection lemma
• epsilon-nets