Geometric Thickness of Multigraphs is ∃R-Complete

We say that a (multi)graph G=(V,E) has geometric thickness t if there exists a straight-line drawing φ:V→R² 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. In this paper, we settle the computational complexity of Geometric Thickness by showing that it is ∃R-complete already for thickness 57. Moreover, our reduction shows that the problem is ∃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 this paper we answer previous questions on 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 ∃R-complete.