We say that a (multi)graph $G = (V,E)$ has geometric thickness $t$ if there exists a straight-line drawing $\varphi : V \rightarrow \mathbb{R}^2$ and a $t$-coloring of its edges where no two edges sharing a point in their relative interior have the same color. The Geometric Thickness problem asks whether a given multigraph has geometric thickness at most $t$. This problem was shown to be NP-hard for $t=2$ [Durocher, Gethner, and Mondal, CG 2016]. In this paper, we settle the computational complexity of Geometric Thickness by showing that it is $\exists \mathbb{R}$-complete already for thickness $57$. Moreover, our reduction shows that the problem is $\exists \mathbb{R}$-complete for $8280$-planar graphs, where a graph is $k$-planar if it admits a topological drawing with at most $k$ crossings per edge. In the course of our paper, we answer previous questions on the geometric thickness and on other related problems, in particular, that simultaneous graph embeddings of $58$ edge-disjoint graphs and pseudo-segment stretchability with chromatic number $57$ are $\exists \mathbb{R}$-complete.

翻译：我们称（多重）图$G = (V,E)$具有几何厚度$t$，若存在直线绘制$\varphi : V \rightarrow \mathbb{R}^2$及其边的$t$染色，使得任意两条在其相对内部存在公共点的边颜色不同。几何厚度问题询问给定多重图是否具有至多为$t$的几何厚度。该问题已被证明在$t=2$时是NP难的 [Durocher, Gethner, and Mondal, CG 2016]。本文通过证明几何厚度问题在厚度$57$时已是$\exists \mathbb{R}$-完备的，完整解决了其计算复杂度。进一步地，我们的归约表明该问题对$8280$-平面图是$\exists \mathbb{R}$-完备的，其中$k$-平面图定义为每条边至多包含$k$个交叉点的拓扑可绘制图。在本文过程中，我们回答了关于几何厚度及其他相关问题的若干已有疑问，特别证明了$58$个边不相交图的同步嵌入问题以及色数为$57$的伪线段可拉伸性问题均为$\exists \mathbb{R}$-完备的。