Order types are an equivalence relation between point configurations that capture their combinatorial and convexity properties. Let $P$ be a $κ$-colored sequence of $n \ge d+1$ points in general position in $\mathbb{R}^d$. Let $ρ$ be a $κ$-colored order type on $k \le d+1$ points that has positive density on $P$; that is, for some constant $δ>0$, there are $δ\cdot \binom{n}{k}$ $k$-point subsequences of $P$ that have the same order type as $ρ$ and the same color pattern. In this paper we show that there exists a constant $c >0$ (depending only on $d, δ$, $k$ and $κ$) and disjoint subsets $X_1,\dots,X_k$ of $P$, each with at least $c \cdot n$ points, such that for every choice of $k$ points $x_i \in X_i$, $(x_1,\dots,x_k)$ has the same order type and color pattern as $ρ$.

翻译：序类型是点配置之间的一种等价关系，它刻画了点的组合与凸性性质。设$P$为$\mathbb{R}^d$中处于一般位置的$n \ge d+1$个点构成的$\kappa$色序列。设$\rho$为$k \le d+1$个点上的$\kappa$色序类型，且在$P$上具有正密度；即存在常数$\delta>0$，使得$P$中有$\delta\cdot \binom{n}{k}$个$k$点子序列与$\rho$具有相同的序类型和颜色模式。本文证明：存在仅依赖于$d, \delta, k$和$\kappa$的常数$c>0$，以及$P$中互不相交的子集$X_1,\dots,X_k$，每个子集至少包含$c\cdot n$个点，使得对任意选取的$k$个点$x_i \in X_i$， $(x_1,\dots,x_k)$与$\rho$具有相同的序类型和颜色模式。