Red-blue clique partitions and (1-1)-transversals

TL;DR

This paper advances the understanding of red-blue clique partitions related to Gallai's problem on (1-1)-transversals of 2-intervals, providing new conditions for vertex partitioning and counterexamples in geometric set systems.

Contribution

It generalizes previous results by weakening the conditions needed for a vertex partition into monochromatic cliques and provides a counterexample in geometric intersection theory.

Findings

Proved that fewer conditions suffice for red-blue clique partitions.

Constructed a counterexample with six 2-convex sets in the plane.

Extended the understanding of (1-1)-transversals in geometric configurations.

Abstract

Motivated by the problem of Gallai on $(1-1)$-transversals of $2$-intervals, it was proved by the authors in 1969 that if the edges of a complete graph $K$ are colored with red and blue (both colors can appear on an edge) so that there is no monochromatic induced $C_4$ and $C_5$ then the vertices of $K$ can be partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani recently strengthened this by showing that it is enough to assume that there is no induced monochromatic $C_4$ and there is no induced $C_5$ in {\em one of the colors}. Here this is strengthened further, it is enough to assume that there is no monochromatic induced $C_4$ and there is no $K_5$ on which both color classes induce a $C_5$. We also answer a question of Kaiser and Rabinovich, giving an example of six $2$-convex sets in the plane such that any three intersect but there is no $(1-1)$-transversal for them.