Let $ Π(n) $ be the largest number such that for every set $ S $ of $ n $ points in a polygon~$ P $, there always exist two points $ x, y \in S $, where every geodesic disk containing $ x $ and $ y $ contains $ Π(n) $ points of~$ S $. We establish upper and lower bounds for $ Π(n)$, and show that $ \left\lceil \frac{n}{5}\right\rceil+1 \leq Π(n) \leq \left\lceil \frac{n}{4} \right\rceil +1 $. We also show that there always exist two points $x, y\in S$ such that every geodesic disk with $x$ and $y$ on its boundary contains at least $ \frac{n}{7+\sqrt{37}} \approx \left\lceil \frac{n}{13.1} \right\rceil$ points both inside and outside the disk. For the special case where the points of $ S $ are restricted to be the vertices of a geodesically convex polygon we give a tight bound of $\left\lceil \frac{n}{3} \right\rceil + 1$. We provide the same tight bound when we only consider geodesic disks having $ x $ and $ y $ as diametral endpoints. We give upper and lower bounds of $\left\lceil \frac{n}{5} \right\rceil + 1 $ and $\frac{n}{6+\sqrt{26}} \approx \left\lceil \frac{n}{11.1} \right\rceil$, respectively, for the two-colored version of the problem. Finally, for the two-colored variant we show that there always exist two points $x, y\in S$ where $x$ and $y$ have different colors and every geodesic disk with $x$ and $y$ on its boundary contains at least $\left\lceil \frac{n}{27.1}\right\rceil+1$ points both inside and outside the disk.

翻译：设$ Π(n) $为最大整数，使得对任意多边形$P$中$n$个点的集合$S$，总存在两点$x,y\in S$，且每个包含$x$和$y$的测地圆盘都包含$S$中至少$Π(n)$个点。我们建立了$ Π(n) $的上界和下界，并证明$\left\lceil \frac{n}{5}\right\rceil+1 \leq Π(n) \leq \left\lceil \frac{n}{4} \right\rceil +1$。我们还证明了总存在两点$x,y\in S$，使得每个以$x$和$y$为边界点的测地圆盘在圆盘内外至少包含$\frac{n}{7+\sqrt{37}} \approx \left\lceil \frac{n}{13.1} \right\rceil$个点。对于$S$中点仅限于测地凸多边形顶点的特殊情况，我们给出了紧界$\left\lceil \frac{n}{3} \right\rceil + 1$。当仅考虑以$x$和$y$为直径端点的测地圆盘时，我们得到相同的紧界。对于该问题的双色版本，我们分别给出了上界和下界$\left\lceil \frac{n}{5} \right\rceil + 1$和$\frac{n}{6+\sqrt{26}} \approx \left\lceil \frac{n}{11.1} \right\rceil$。最后，对于双色变体，我们证明了总存在两个颜色不同的点$x,y\in S$，使得每个以$x$和$y$为边界点的测地圆盘在圆盘内外至少包含$\left\lceil \frac{n}{27.1}\right\rceil+1$个点。