No abstract

In 1958 B. Griinbaum made a conjecture concerning families of disjoint translates of a compact convex set in the plane: if such a family consists of at least five sets, and if any five of these sets are met by a common line, then some line meets all sets of the family. This paper gives a proof of the conjecture.

A short proof is given of the impossibility of decomposing the complete graph on n vertices into n-2 or fewer complete bipartite graphs.Let G be a finite graph, and let V , , . . . , V, be sets of vertices of G. Assume that for each i, Gi is a subgraph of G( VJ, the subgraph induced by G on Vi .Assume furthermore that the Gj are edgedisjoint and between them contain all edges of G. Then GI, . . . , G, form a decomposition of G. It was proved in[ I ] that if G = K,, the complete graph on n vertices, and each Gj is a complete bipartite graph, then r 1 n -1. This inequality is a consequence of a theorem in [ 11, and in [2] it is remarked that the application of that theorem still seems to be the only known way of proving it (a similar remark was made in [ 11). Below I give a direct proof. I thank K.P. Villanger for bringing the problem to my attention.

For d > 2 and n > d + 1, let P = {p 1 , . . . , p n } be a set of points in R d whose convex hull contains the origin O in its interior. We show that if P ∪ {O} is in general position, then there exists a d-tuple Q = {p i 1 , . . . , p i d } ⊂ P such that O is not contained in the convex hull of Q ∪ {p} for any p ∈ P \ Q. Generalizations of this property are also considered.We also show that for non-empty, finite point sets A 1 , . . . , A d+1 in R d , if the origin is contained in the convex hull of A i ∪A j for all 1 ≤ i < j ≤ d +1, then there is a simplex S containing the origin such that |S ∩ A i | = 1 for every 1 ≤ i ≤ d + 1. This is a generalization of Bárány's colored Carathéodory theorem, and dually, it gives a spherical version of Lovász' colored Helly theorem.

No abstract