We study $\mu_5(n)$, the minimum number of convex pentagons induced by $n$ points in the plane in general position. Despite a significant body of research in understanding $\mu_4(n)$, the variant concerning convex quadrilaterals, not much is known about $\mu_5(n)$. We present two explicit constructions, inspired by point placements obtained through a combination of Stochastic Local Search and a program for realizability of point sets, that provide $\mu_5(n) \leq \binom{\lfloor n/2 \rfloor}{5} + \binom{\lceil n/2 \rceil}{5}$. Furthermore, we conjecture this bound to be optimal, and provide partial evidence by leveraging a MaxSAT encoding that allows us to verify our conjecture for $n \leq 16$.

翻译：我们研究 $\mu_5(n)$，即处于一般位置的平面 $n$ 个点所诱导的凸五边形的最小数目。尽管关于 $\mu_4(n)$（凸四边形的变体）已有大量研究，但对 $\mu_5(n)$ 的了解仍然有限。我们提出两种显式构造方法，这些方法灵感来源于结合随机局部搜索与点集可实现性程序获得的点布局，并给出 $\mu_5(n) \leq \binom{\lfloor n/2 \rfloor}{5} + \binom{\lceil n/2 \rceil}{5}$ 的结果。此外，我们推测该上界是最优的，并通过利用 MaxSAT 编码为 $n \leq 16$ 的情形验证了该猜想，从而提供了部分证据。