In this short paper, 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 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. Our results are a generalization of $(\aleph_0,k+2)$-Theorem for $k$-transversals of convex sets by Keller and Perles (Symposium on Computational Geometry 2022), and can also be seen as a colorful infinite variant of $(p,q)$-Theorems of Alon and Klietman (Advances in Mathematics 1992), and Alon and Kalai (Discrete & Computational Geometry 1995).

翻译：在本文中，我们证明：若$\{\mathcal{F}_{n}\}_{n \in \mathbb{N}}$是$\mathbb{R}^{d}$中一族紧致$(r,R)$-胖凸集，且每个关于$\{\mathcal{F}_{n}\}_{n \in \mathbb{N}}$的异色序列均包含$k+2$个可被同一$k$-平面刺穿的凸集，则存在该族中的某个子集$\mathcal{F}_{m}$可被有限个$k$-平面刺穿。此外，我们证明：若$\{\mathcal{F}_{n}\}_{n \in \mathbb{N}}$是$\mathbb{R}^{d}$中一族紧致凸集（每个$\mathcal{F}_{n}$均为$\mathbb{R}^{d}$中的闭球或轴平行盒），且每个关于$\{\mathcal{F}_{n}\}_{n \in \mathbb{N}}$的异色序列均包含两个相交的闭球（或盒），则存在该族中的某个子集$\mathcal{F}_{m}$可被$\mathbb{R}^{d}$中有限个点刺穿。为补充上述结果，我们还证明了对于其他更一般的凸集族，类似结论无法成立。本文结果推广了Keller与Perles（计算几何研讨会, 2022）关于凸集$k$-横截的$(\aleph_0,k+2)$定理，并可作为Alon与Kleitman（数学进展, 1992）及Alon与Kalai（离散与计算几何, 1995）的$(p,q)$定理的彩色无穷变体。