Motivated by the problem of Gallai on (1â1)-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 C4â and C5â 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 C4â and there is no induced C5â in {\em one of the
colors}. Here this is strengthened further, it is enough to assume that there
is no monochromatic induced C4â and there is no K5â on which both color
classes induce a C5â.
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
(1â1)-transversal for them