Independent sets in K4-free graphs

Prove that every K4-free graph with maximum degree Δ on n vertices contains an independent set of size at least c n log Δ/Δ for some absolute constant c>0.

Background

For K4-free graphs, every vertex neighbourhood induces a triangle-free graph, but neighbourhoods may still have substantial edge density. The paper states that the best currently available bound is of order n log Δ/(Δ log log Δ), whereas the longstanding Ajtai–Erdős–Komlós–Szemerédi conjecture predicts removal of the extra log log Δ factor. Establishing the conjectured bound would substantially improve the known independence-number estimate for K4-free bounded-degree graphs.

References

There is some distance to cover, as the longstanding conjecture of Ajtai, Erd\H{o}s, Koml\'os and Szemer\'edi posits much more.

— The hard-core model in graph theory  (2501.03379 - Davies et al., 6 Jan 2025) in Conjecture C, Section “Open problems”