“…These observations have led to a number of interesting results and conjectures concerning families satisfying the colorful intersection property [3,4,5]. Here we focus on the remarkable results by Montejano and Karasev [5], which state that under certain conditions on n, |F i | and d, the colorful intersection property implies that that one of the families F i admits a k-transversal, that is, a k-dimensional affine flat that intersects every memebr of F i . In its simplest non-trivial form, their theorem can be formulated as follows (See also [4,Theorem 3.2]): Consider 3 red convex sets and 3 blue convex sets in R 3 , and suppose every red set intersects every blue set.…”