Let $P$ be a $k$-colored set of $n$ points in the plane, $4 \leq k \leq n$. We study the problem of deciding if $P$ contains a subset of four points of different colors such that its Rectilinear Convex Hull has positive area. We provide an $O(n \log n)$-time algorithm for this problem, where the hidden constant does not depend on $k$; then, we prove that this problem has time complexity $\Omega(n \log n)$ in the algebraic computation tree model. No general position assumptions for $P$ are required.

翻译：设$P$为平面上$n$个点构成的$k$-着色集，其中$4 \leq k \leq n$。我们研究判定$P$是否包含四个不同颜色的点，使其直线凸包具有正面积的问题。针对该问题，我们提出一个$O(n \log n)$时间的算法，其中隐藏常数不依赖于$k$；随后，我们证明该问题在代数计算树模型下的时间复杂度为$\Omega(n \log n)$。无需对$P$作一般位置假设。