Red-Blue Clique Partitions and (1-1)-Transversals
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 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.