Given any integers s,t≥2, we show there exists some c=c(s,t)>0 such
that any Ks,t-free graph with average degree d contains a subdivision of
a clique with at least cd21s−1s vertices. In particular,
when s=2 this resolves in a strong sense the conjecture of Mader in 1999 that
every C4-free graph has a subdivision of a clique with order linear in the
average degree of the original graph. In general, the widely conjectured
asymptotic behaviour of the extremal density of Ks,t-free graphs suggests
our result is tight up to the constant c(s,t).Comment: 25 pages, 1 figur