Let $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be an infinite collection of families of compact convex sets in $\mathbb{R}^{d}$. An infinite sequence of compact convex sets $\{B_n\}_{n\in\mathbb{N}}$ is said to be a heterochromatic sequence with respect to $\left\{ \mathcal{F}_{n}\right\}_{n\in \mathbb{N}}$ if each $B_n$ comes from a different family $\mathcal{F}_{i_n}$, and $\{B_n\}_{n\in\mathbb{N}}$ is said to be a strongly heterochromatic sequence of $\left\{ \mathcal{F}_{n}\right\}_{n\in \mathbb{N}}$ if $\forall n\in\mathbb{N}, \; B_n \in \mathcal{F}_n$. We show that if $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be a collection of families compact $(r, R)$-fat convex sets in $\mathbb{R}^{d}$ and if every heterochromatic sequence with respect to $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ contains $k+2$ convex sets that can be pierced by a $k$-flat then there exists a family $\mathcal{F}_{m}$ from the collection that can be pierced by finitely many $k$-flats. Additionally, we show that if $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be a collection of families of compact convex sets in $\mathbb{R}^{d}$ where each $\mathcal{F}_{n}$ is a family of closed balls (axis parallel boxes) in $\mathbb{R}^{d}$ and every (strongly) heterochromatic sequence with respect to $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ contains $2$ intersecting closed balls (boxes) then there exists a family $\mathcal{F}_{m}$ from the collection that can be pierced by a finite number of points from $\mathbb{R}^{d}$. To complement the above results, we also establish some impossibility of proving similar results for other more general families of convex sets.

翻译：设 $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ 是 $\mathbb{R}^{d}$ 中一族紧凸集族构成的无限集合。若每个 $B_n$ 来自不同族 $\mathcal{F}_{i_n}$，则称无限序列 $\{B_n\}_{n\in\mathbb{N}}$ 为关于 $\left\{ \mathcal{F}_{n}\right\}_{n\in \mathbb{N}}$ 的异色序列；若 $\forall n\in\mathbb{N}, \; B_n \in \mathcal{F}_n$，则称 $\{B_n\}_{n\in\mathbb{N}}$ 为 $\left\{ \mathcal{F}_{n}\right\}_{n\in \mathbb{N}}$ 的强异色序列。我们证明：若 $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ 是 $\mathbb{R}^{d}$ 中一族紧 $(r, R)$-胖凸集族，且关于 $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ 的每个异色序列都包含 $k+2$ 个可被 $k$-平面刺穿的凸集，则存在该集合中的某个族 $\mathcal{F}_{m}$ 可被有限个 $k$-平面刺穿。此外，我们证明：若 $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ 是 $\mathbb{R}^{d}$ 中一族紧凸集族，其中每个 $\mathcal{F}_{n}$ 为 $\mathbb{R}^{d}$ 中的闭球族（轴平行盒族），且关于 $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ 的每个（强）异色序列都包含 $2$ 个相交的闭球（盒），则存在该集合中的某个族 $\mathcal{F}_{m}$ 可被 $\mathbb{R}^{d}$ 中有限个点刺穿。为补充上述结果，我们还证明对于其他更一般的凸集族，类似结果无法成立。