This paper is concerned with the linkedness of Cartesian products complete graphs. A graph at least 2 k vertices -linked if, for every set distinct organised in arbitrary pairs vertices, there are vertex-disjoint paths joining pairs. We show that product K d 1 +1 × graphs and ⌊( + )/2⌋ , ≥ this best possible. result connected to simple polytopes. The T ( ) a -dimensional simplex . And polytope ...