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

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  • Warwick University


Given any integers s,t2, 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 cds/2(s1) 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,‐free graphs suggests our result is tight up to the constant c(s,t).


Original languageEnglish
Pages (from-to)203-222
Number of pages20
JournalJournal of the London Mathematical Society
Issue number1
Early online date6 Jan 2017
Publication statusPublished - Feb 2017


  • 05C35 (primary), 05C38, 05C83 (secondary)