Subdivisions of a large clique in C6-free graphs

Mader conjectured that every C4-free graph has a subdivision of a clique of order linear in its average degree. We show that every C6-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 C4-free graph with average degree cn 1/2 has a subdivision of a clique K` with ` = bc′n1/2c where every edge is subdivided exactly 3 times.