An infinite sequence of sets $\left\{B_{n}\right\}_{n\in\mathbb{N}}$ is said to be a heterochromatic sequence from an infinite sequence of families $\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}$, if there exists a strictly increasing sequence of natural numbers $\left\{ i_{n}\right\}_{n \in \mathbb{N}}$ such that for all $n \in \mathbb{N}$ we have $B_{n} \in \mathcal{F}_{i_{n}}$. In this paper, we have proved that if for each $n\in\mathbb{N}$, $\mathcal{F}_n$ is a family of {\em nicely shaped} convex sets in $\mathbb{R}^d$ such that each heterochromatic sequence $\left\{B_{n}\right\}_{n\in\mathbb{N}}$ from $\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}$ contains at least $k+2$ sets that can be pierced by a single $k$-flat ($k$-dimensional affine space) then all but finitely many families in $\left\{\mathcal{F}_{n}\right\}_{n\in \mathbb{N}}$ can be pierced by finitely many $k$-flats. This result can be considered as a {\em countably colorful} generalization of the $(\aleph_0, k+2)$-theorem proved by Keller and Perles (Symposium on Computational Geometry 2022). We have also established the tightness of our result by proving a number of no-go theorems.

翻译：设$\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}$为无限个集合族构成的序列。若存在严格递增的自然数序列$\left\{ i_{n}\right\}_{n \in \mathbb{N}}$使得对所有$n \in \mathbb{N}$均有$B_{n} \in \mathcal{F}_{i_{n}}$，则称无限集合序列$\left\{B_{n}\right\}_{n\in\mathbb{N}}$为$\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}$的异色序列。本文证明：若对每个$n\in\mathbb{N}$，$\mathcal{F}_n$是$\mathbb{R}^d$中一族“形状良好”的凸集，且从$\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}$中选取的每个异色序列$\left\{B_{n}\right\}_{n\in\mathbb{N}}$均包含至少$k+2$个能被同一$k$-平面（$k$维仿射空间）穿透的集合，则$\left\{\mathcal{F}_{n}\right\}_{n\in \mathbb{N}}$中除有限个族外，其余均可被有限个$k$-平面穿透。该结果可视为Keller与Perles（计算几何研讨会2022）提出的$(\aleph_0, k+2)$-定理的“可数彩色”推广。通过证明一系列否定性定理，我们进一步确立了结果的紧性。