Subdivisions and near-linear stable sets

TL;DR

This paper proves that graphs avoiding subdivisions of complete graphs contain large stable sets, nearly matching the best possible bounds, with implications for understanding graph structure and stability properties.

Contribution

It establishes near-linear lower bounds on the size of stable sets in graphs excluding subdivisions of complete graphs, advancing knowledge on graph stability and structure.

Findings

Graphs with no subdivision of $K_t$ have stable sets of size at least $|G|/polylog|G|$

For $t \\ge 7$, such graphs may not have linear-sized stable sets

The results are close to optimal bounds for these classes of graphs

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.