On cliques in diameter graphs

Note that we assume of the sphere being embedded in R, and the unit distance included from the ambient space. Diameter graphs arise naturally in the context of Borsuk’s problem. In 1933 Borsuk [3] asked whether any set of diameter 1 in R can be partitioned into (d+1) parts of strictly smaller diameter. The positive answer to this question is called Borsuk’s conjecture. This was shown to be true in dimensions up to 3. In 1993 Kahn and Kalai [6] constructed a finite set of points in dimensions 1325 that does not admit a partition into 1326 parts of smaller diameter. The minimal dimension in which the counterexample is known is 64 (see [2], [5]). We focus on one conjecture, posed by Morić and Pach [12].