On balanced 4-holes in bichromatic point sets

Let S = R∪B be a set of n points in general position in the plane. The elements of R and B will be called, respectively, the red and blue elements of S. A k-hole of S is a simple polygon with k vertices, all in S, and containing no element of S in its interior. A 4-hole of S is balanced if it has two blue and two red vertices. In this paper, we characterize the set of bicolored points S = R∪B that have balanced convex 4-holes. We also show that if the 4-holes of S are allowed to be nonconvex, and |R| = |B|, then S always has a quadratic number of balanced 4-holes. The study of k-holes in colored point sets was introduced by Devillers et al. [2]. They obtained a bichromatic point set S = R ∪B with 18 points that contains no convex monochromatic 4-hole. Recently, Hummer and Seara obtained a bichromatic point set with 36 points that does not contain monochromatic 4-holes [3]. This result was improved by Koshelev [4] to 46. Devillers et al. [2] also proved that every 2colored Horton set with at least 64 elements contains an empty monochromatic 4-hole. In the same paper the following conjecture is posed: Every sufficiently large bichromatic point set contains a monochromatic convex 4-hole. This conjecture remains open. On the other hand, Aichholzer et al [1] proved that any sufficiently large bichromatic point set always contains a not necessarily convex monochromatic 4-hole.