Universal extremal bounds for fixed-dimensional box-Delaunay graphs and Hasse diagrams

Determine whether every n-vertex box-Delaunay graph or Hasse diagram of each fixed dimension d≥2 has chromatic number at most (log n)^{d−1+o(1)} and independence number at least n/(log n)^{d−1+o(1)}; in particular, determine whether every two-dimensional such graph has chromatic number n^{o(1)} and independence number n^{1−o(1)}.

Background

The main theorems describe the typical behavior of graphs generated by uniformly random point sets, while this problem asks for corresponding worst-case bounds over all n-vertex graphs in the same geometric and order-theoretic classes.

The authors state that the conjectured asymptotic extremal behavior currently seems far out of reach and single out the two-dimensional subquestions as especially important open questions. These would substantially strengthen the known results on fixed-dimensional Hasse diagrams and box-Delaunay graphs.

References

Actually, one might wonder whether (for every constant d\ge2) $(\log n){d-1+o(1)}$ is the maximum chromatic number of any $n$-vertex box-Delaunay graph or Hasse diagram of dimension $d$, and $n/(\log n){d-1+o(1)}$ is the minimum independence number. However, this currently seems far out of reach: some of the most important open questions in this direction are whether the chromatic number of a 2-dimensional Hasse diagram or box-Delaunay graph is always $n{o(1)}$, and whether the independence number is always $n{1-o(1)}$.

— Colouring random Hasse diagrams and box-Delaunay graphs  (2501.12373 - Jin et al., 21 Jan 2025) in Section 1, subsection “Further directions”