How do 9 points look like in $E^3$?

Abstract

The aim of this note is to give an elementary proof of the following fact: given 3 red convex sets and 3 blue convex sets in ${\mathbb{E}}^3$, such that every red intersects every blue, there is a line transversal to the reds or there is a line transversal to the blues. This is a special case of a theorem of Montajano and Karasev \cite{MK} and generalizes, in a sense, the colourful Helly theorem due to Lovász (cf. \cite{BL}).

To appear in Annals of Combinatorics