Counterexamples to the Colorful Tverberg Conjecture for Hyperplanes

TL;DR

This paper disproves Karasev's 2008 conjecture that certain line configurations always admit intersecting colorful triples, providing counterexamples in the plane and higher dimensions.

Contribution

The authors construct counterexamples to the Colorful Tverberg Conjecture for hyperplanes, disproving its universal validity.

Findings

Counterexamples exist for all r in the plane.

Counterexamples extend to higher-dimensional spaces.

The conjecture does not hold universally for line and hyperplane arrangements.

Abstract

In 2008 Karasev conjectured that for every set of $r$ blue lines, $r$ green lines, and $r$ red lines in the plane, there exists a partition of them into $r$ colorful triples whose induced triangles intersect. We disprove this conjecture for every $r$ and extend the counterexamples to higher dimensions.