A proof of Mader's conjecture on large clique subdivisions in C4-free graphs

Given any integers s, t >= 2, we show that 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 cd(s/2(s-1)) vertices. In particular, when s = 2, this resolves in a strong sense the conjecture of Mader in 1999 that every C-4-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 K-s,K-t-free graphs suggests our result is tight up to the constant c(s, t).