Induced subdivisions in graphs of large girth

Abstract

In this paper, we prove that there exists an absolute constant $g_0$ such that, for every integer $k\ge 3$, every graph $G$ with $\delta (G)\ge k$ and $g(G)\ge g_0$ contains an induced subdivision of $K_{k+1}$. This fully resolves a problem raised by Kühn and Osthus (originally attributed to Shi), and improves a recent result of Girão and Hunter. Our proof uses some ideas from Girão and Hunter. Another main ingredient in our proof is an induced variant of Mader's theorem: for every fixed \(s,η,D\), every graph \(J\) with \(Δ(J)\le D\), \(d(J)>s-2+η\) and sufficiently large girth contains an induced subdivision of \(K_s\).