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：若存在严格递增的自然数序列 $\left\{ i_{n}\right\}_{n \in \mathbb{N}}$，使得对于所有 $n \in \mathbb{N}$ 均有 $B_{n} \in \mathcal{F}_{i_{n}}$，则称无穷集合序列 $\left\{B_{n}\right\}_{n\in\mathbb{N}}$ 为无穷族序列 $\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}$ 的异色序列。本文证明：若对每个 $n\in\mathbb{N}$，$\mathcal{F}_n$ 是 $\mathbb{R}^d$ 中一族具有良好几何形状的凸集，且每个来自 $\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}$ 的异色序列 $\left\{B_{n}\right\}_{n\in\mathbb{N}}$ 都包含至少 $k+2$ 个能被同一 $k$-平坦（$k$ 维仿射空间）贯穿的集合，则 $\left\{\mathcal{F}_{n}\right\}_{n\in \mathbb{N}}$ 中除有限个族外，均可被有限个 $k$-平坦贯穿。该结果可视为 Keller 与 Perles（计算几何研讨会 2022）所证明的 $(\aleph_0, k+2)$ 定理的可数多彩推广。我们还通过证明若干不可行定理，确立了该结果的紧致性。