Subdivisions and near-linear stable sets

TL;DR

The paper addresses the problem of obtaining large stable sets in graphs that exclude subdivisions of a fixed graph, proving that for every $t\ge3$ any $K_t$-subdivision-free graph $G$ has $\alpha(G) \ge c_t|G|/(\log|G|)^{3t-5}$ with $c_t=(2t)^{-4(t-2)}$. The authors develop a two-phase inductive framework that (i) builds $t$ nonadjacent high-degree vertices with suitably sparse neighbor sets to form $t$ stars, and (ii) routes induced paths between these neighbor sets to realize a subdivision of $K_t$ while carefully controlling disqualified vertices via expansion arguments and a logarithmic degree factor. The main contribution is a near-linear lower bound on $\alpha(G)$ under subdivision-free constraints, which extends existing results for geometric graph classes and yields polylogarithmic bounds on chromatic numbers; it also clarifies the relationship between excluding forests and excluding subdivisions, and suggests a unified perspective through $H(T)$-subdivision-free graphs. The work has implications for related conjectures (e.g., Gyárfás–Sumner) and strengthens the understanding of how induced subdivisions constrain the structure of graphs with bounded clique number.

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 $|G|/{\rm polylog}|G|$. 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.