  A theorem of Mader shows that every graph with average degree at least eight
has a $K_6$ minor, and this is false if we replace eight by any smaller
constant. Replacing average degree by minimum degree seems to make little
difference: we do not know whether all graphs with minimum degree at least
seven have $K_6$ minors, but minimum degree six is certainly not enough. For
every $c>0$ there are arbitrarily large graphs with average degree at least
$8-c$ and minimum degree at least six, with no $K_6$ minor.
  But what if we restrict ourselves to bipartite graphs? The first statement
remains true: for every $c>0$ there are arbitrarily large bipartite graphs with
average degree at least $8-c$ and no $K_6$ minor. But surprisingly, going to
minimum degree now makes a significant difference. We will show that every
bipartite graph with minimum degree at least six has a $K_6$ minor. Indeed, it
is enough that every vertex in the larger part of the bipartition has degree at
least six.
