Subdivisions and near-linear stable sets

Tung Nguyen, Alex Scott, Paul Seymour

Abstract: 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 \(|G|/\operatorname {polylog}|G|\). 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.