We consider colored variants of a class of geometric-combinatorial questions on $k$-gons and empty $k$-gons that have been started around 1935 by Erdős and Szekeres. In our setting we have $n$ points in general position in the plane, each one colored either red or blue. A structure on $k$ points is a geometric graph where the edges are spanned by (some of) these points and is called monochromatic if all $k$ points have the same color. Already for $k=4$ there exist interesting open problems. Most prominently, it is still open whether for any sufficiently large bichromatic set there always exists a convex empty, monochromatic quadrilateral. In order to shed more light on the underlying geometry we study the existence of five different monochromatic structures that all use exactly 4 points of a bichromatic point set. We provide several improved lower and upper bounds on the smallest $n$ such that every bichromatic set of at least $n$ points contains (some of) those monochromatic structures.

翻译：我们考虑一类始于1935年由Erdős和Szekeres提出的关于$k$边形与空$k$边形的几何组合问题的彩色变体。在我们的设定中，平面上有$n$个处于一般位置的点，每个点被染成红色或蓝色。一个包含$k$个点的结构是由这些点（或其子集）张成的几何图，若所有$k$个点颜色相同，则称该结构为单色的。即使对于$k=4$，也已存在有趣的开放问题。最突出的问题是：对于任意足够大的双色点集，是否总存在一个凸的、空的、单色的四边形？为了更深入地理解其背后的几何性质，我们研究了恰好使用双色点集中4个点的五种不同单色结构的存在性。我们针对使得任意至少包含$n$个点的双色点集必然包含（某些）此类单色结构的最小$n$值，给出了若干改进的下界和上界。