Improved Bounds for Topological Cliques in Graphs of Large Girth
Daniela Kühn, Deryk Osthus · SIAM Journal on Discrete Mathematics · 2006
We prove that every graph of minimum degree at least r and girth at least 27 contains a subdivision of $K_{r+1}$. This implies that the conjecture of Hajós, that every graph of chromatic number at least r contains a subdivision of $K_r$, is true for graphs of girth at least 27. This conjecture is known to be false in general.