On a conjecture of Karasev

TL;DR

This paper investigates a geometric conjecture about partitioning lines into colorful triangles with a common intersection point, proving it holds for lines in convex position despite being false in general.

Contribution

The paper proves Karasev's conjecture for lines in convex position, providing a specific case where the conjecture holds and exploring potential generalizations.

Findings

Karasev's conjecture is false in general.

The conjecture holds for lines in convex position.

Discussion of possible extensions to other configurations.

Abstract

Karasev conjectured that for any set of $3k$ lines in general position in the plane, which is partitioned into $3$ color classes of equal size $k$, the set can be partitioned into $k$ colorful 3-subsets such that all the triangles formed by the subsets have a point in common. Although the general conjecture is false, we show that Karasev's conjecture is true for lines in convex position. We also discuss possible generalizations of this result.