Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let $\operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $\operatorname{rb-index}(k)$ be the maximum of $\operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $\operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $\operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $\operatorname{rb-index}(k)\neq k$, and we prove that for $k\ge 5$, \[ \frac{40\lfloor (k-1)/2 \rfloor -8}{19} %Birgit: \leq\operatorname{rb-index}(k)\leq 10 \bigg\lfloor\frac{k}{7}\bigg\rfloor + 11. \] Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 \lfloor\frac{k}{7}\rfloor + 11$ vertices can be computed in $O(n\log n)$ time.

翻译：根据平面上设定的彩虹点,完美的彩虹多边形是一个简单的多边形, 包含每个颜色的一点, 不管是内部还是边界。 $\operatorname{rb- ind} (S) 表示彩虹多边形最小大小, 彩色点上设定了$S$, 并且$\operatorname{rb- ind} (k) 美元是$\opatorname{rb- index} (S) 的最大值, 在所有的 $k$(k) 的平面点上 ; 也就是说, 每张彩色点上设定的$(k) $(k) $(c) $(c) $(c) $(r) $(r_r- indior_ leg_ leg_ leg_ leg_ dir_ leg_ dir_ leg_ dir_ leg_ g_r_ dir_ dir_ leg_ dal_ dal_ leg_ dal_ dal_ dal_ dal_ dal_ dal_ dal_ g_ dir_ dal_ d_ dir_ d_ d_ d_ g_ dir_ dir_ d_ d_ d_ d_ dir_ lexxx_ d_r_r_ g_____ d_r_r_r__________ lex____ lex_r_____ lex_ d_ d_ d_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_r_l___l_l__