Induced subdivisions in graphs of large girth

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 $s,\eta ,D$, every graph $J$ with $\mathrm{\Delta }(J)\le D$, $d(J)>s-2+\eta$ and sufficiently large girth contains an induced subdivision of $K_s$.