  Mader conjectured that every $C_4$-free graph has a subdivision of a clique
of order linear in its average degree. We show that every $C_6$-free graph has
such a subdivision of a large clique.
  We also prove the dense case of Mader's conjecture in a stronger sense, i.e.
for every $c$, there is a $c'$ such that every $C_4$-free graph with average
degree $cn^{1/2}$ has a subdivision of a clique $K_\ell$ with $\ell=\lfloor
c'n^{1/2}\rfloor$ where every edge is subdivided exactly $3$ times.
