Improving Johansson’s bound for K_T-free graphs

Prove whether the chromatic-number bound O_T(Δ log log Δ/log Δ) for K_T-free graphs of maximum degree Δ can be improved by a factor of log log Δ.

Background

The paper uses Johansson’s theorem for K_T-free graphs to control the chromatic number of box-Delaunay graphs, but that theorem introduces an additional log log factor. A stronger bound would improve the general graph-theoretic tool available for the coloring argument.

The authors state that such an improvement is conjectured but remains unresolved; they therefore use a different theorem of Alon, Krivelevich, and Sudakov to obtain the precise bound needed for their random graph results.

References

It is conjectured that Johansson's bound for $K_{T}$-free graphs can be improved by a factor of $\log\log \Delta$ but, as this is still an open problem, we need to find another route to prove the precise bound in \cref{eq:chi}.

— Colouring random Hasse diagrams and box-Delaunay graphs  (2501.12373 - Jin et al., 21 Jan 2025) in Section 2, subsection “Upper bounds on the chromatic number”