On a conjecture of Karasev

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.