Pushed colorful polytopal KKMS Theorem and orthogonal star transversals

A classical theorem of Kaiser and Rabinovich implies that if every subfamily of size at most $\lfloor {\log }_2(d+1)\rfloor +1$ of a finite family of compact connected sets in ${\mathbb{R}}^d$ has a common axis-parallel hyperplane transversal, then the entire family has an orthogonal star transversal, i.e., a union of $d$ hyperplanes, one perpendicular to each coordinate axis. We study colorful variants of this result. First, we use standard KKM machinery to establish a colorful analogue with $d+1$ color classes and a $\lfloor {\log }_2(d+1)\rfloor +1$ transversal hypothesis for rainbow subfamiles. Then we focus on reducing the required number of colors. To this end, we introduce the rainbow pushability ${\pi }_r(P)$ of a polytope and we use it to state and prove few-color variants of the colorful Carathéodory theorem and the colorful polytopal KKMS theorem. Applying the latter to the $d$-dimensional cube, we show that $\lfloor {\log }_{2}d\rfloor +2$ color classes suffice for our transversal problem, provided that every rainbow choice has a common axis-parallel hyperplane transversal. We also extend both colorful results to the setting of convex $(n,D)$-bodies and we show that three colors are needed for $d\ge 2$.