  This paper is concerned with the linkedness of Cartesian products of complete
graphs. A graph with at least $2k$ vertices is {\it $k$-linked} if, for every
set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices,
there are $k$ vertex-disjoint paths joining the vertices in the pairs.
  We show that the Cartesian product $K^{d_{1}+1}\times K^{d_{2}+1}$ of
complete graphs $K^{d_{1}+1}$ and $K^{d_{2}+1}$ is
$\floor{(d_{1}+d_{2})/2}$-linked for $d_{1},d_{2}\ge 2$, and this is best
possible.
  %A polytope is said to be {\it $k$-linked} if its graph is $k$-linked.
  This result is connected to graphs of simple polytopes. The Cartesian product
$K^{d_{1}+1}\times K^{d_{2}+1}$ is the graph of the Cartesian product
$T(d_{1})\times T(d_{2})$ of a $d_{1}$-dimensional simplex $T(d_{1})$ and a
$d_{2}$-dimensional simplex $T(d_{2})$. And the polytope $T(d_{1})\times
T(d_{2})$ is a {\it simple polytope}, a $(d_{1}+d_{2})$-dimensional polytope in
which every vertex is incident to exactly $d_{1}+d_{2}$ edges.
  While not every $d$-polytope is $\floor{d/2}$-linked, it may be conjectured
that every simple $d$-polytope is. Our result implies the veracity of the
revised conjecture for Cartesian products of two simplices.
