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

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 cd 1 2 s s−1 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… (More)