Distance graphs in R

Analogously, we may define a diameter graph on the sphere S r of radius r. We think of the sphere being embedded in R, and the (unit) distance is induced from the ambient space. Diameter graphs are closely related to the famous Borsuk problem. In 1933 Borsuk [1] asked, whether any set of diameter 1 in R can be partitioned into (d+ 1) parts of strictly smaller diameter. The positive answer to the question is known as Borsuk’s conjecture. This was shown to be true in dimensions up to 3. For 60 years the question in higher dimensions remained open, until in 1993 Kahn and Kalai [7] constructed a finite set of points in dimension 1325 that does not admit a partition into 1326 parts of smaller diameter. The minimal dimension in which the counterexample is known was reduced by several authors, with a current record d = 65 due to Bondarenko. Borsuk’s problem for finite sets in R can be formulated in terms of diameter graphs. Namely, whether it is true that any diameter graph G in R satisfies χ(G) 6 d+1? This and related problems were studied by several authors. In [6] Hopf and Pannwitz proved that the number of edges in any diameter graph in R is at most n, which easily implies Borsuk’s conjecture for finite sets on the plane. Vázsonyi conjectured, that any diameter graph in R on n vertices can have at most 2n − 2 edges. Again, it is not difficult to see that Vázsonyi’s conjecture implies Borsuk’s conjecture for finite sets in R. Vázsonyi’s conjecture was proved independently by Grünbaum [4], Heppes [5] and Straszewicz [10]. Almost 50 years later, two other papers on this topic appeared. In [2] Dol’nikov proved that in a diameter graph in R any two odd cycles must share a vertex. Out of this statement he derived Borsuk’s conjecture for finite sets. The method he introduced was later developed by Swanepoel [12], who managed to give yet another proof of Vázsonyi’s conjecture.