Rectangular grids in crossing segment arrangements

For every integer \(k\geq1\), an arrangement of \(3k\) disjoint red straight-line segments and \(3k\) disjoint blue straight-line segments, with every red--blue pair crossing in both relative interiors, contains \(k\) segments of each color forming an embedded rectangular grid. This answers Question 1 of Jain, Ricci, Rollin and Schulz. The equivalence preserves the entire subdivided segment paths, their cyclic orders and the unbounded face. More generally, \(n\) segments of each color contain a rectangular subarrangement of \(p\) red and \(n-p+1\) blue segments for every \(1\leq p\leq n\); consequently \(2k-1\) segments of each color suffice for a \(k\)-grid. The selection follows by averaging consecutive gaps in the circular order of the segment directions.