Subdivisions and near-linear stable sets

We prove that for every complete graph $K_t$, all graphs $G$ with no induced subgraph isomorphic to a subdivision of $K_t$ have a stable subset of size at least $\mathrm{\mid }G\mathrm{\mid }\mathrm{/}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{\mid }G\mathrm{\mid }$. This is close to best possible, because for $t\ge 7$, not all such graphs $G$ have a stable set of linear size, even if $G$ is triangle-free.