Let S be a set of n points in general position in the plane. Suppose that each point of S has been assigned one of $k\ge 3$ possible colors and that there is the same number, m, of points of each color class. This means $n=km$. A polygon with vertices on S is empty if it does not contain points of S in its interior; and it is rainbow if all its vertices have different colors. Let $f(k,m)$ be the minimum number of empty rainbow triangles determined by S. In this paper we give tight asymptotic bounds for this function. Furthermore, we show that S may not determine an empty rainbow quadrilateral for some arbitrarily large values of k and m.

中文翻译：

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K色点集中的空彩虹三角形

设S为平面中一般位置上的n个点的集合。假设已为S的每个点分配了一个$ķ\ge 3$可能的颜色，并且每种颜色类别的点数相同m。这意味着$ñ=ķ米$。如果在其内部不包含S的点，则在S上具有顶点的多边形为空。如果所有顶点的颜色都不同，那就是彩虹。让$F（ķ，米）$是由S确定的最小彩虹三角形的最小数量。在本文中，我们为此函数给出了严格的渐近界。此外，我们表明对于某些任意大的k和m值，S可能无法确定空彩虹四边形。