Combining the higher-dimensional and two-dimensional exposure methods

Determine whether the ideas based on the two-dimensional short-interval analysis can be combined with the higher-dimensional inductive suitable-pair method in order to close the remaining bounds gap for random Hasse diagrams and box-Delaunay graphs.

Background

The higher-dimensional proof uses only the randomness of marked points to overcome dependence issues, whereas the sharper two-dimensional analysis exploits fluctuations involving the unmarked points. The authors explain that these two sources of randomness lead to technically different approaches.

The paper explicitly leaves unresolved whether the two methods can be integrated. Such a combination is identified as likely necessary to obtain the conjecturally optimal higher-dimensional bounds.

References

We remark that the improvement for the $d=2$ case described in \cref{subsubsec:sharper} seems to fundamentally depend on the randomness of the unmarked points, so it is not clear to us how to combine the ideas in this subsection with the ideas in \cref{subsubsec:sharper}. We believe that closing the gap in \cref{thm:d>2} would require combining ideas from both approaches.

— Colouring random Hasse diagrams and box-Delaunay graphs  (2501.12373 - Jin et al., 21 Jan 2025) in Section 2, subsection “Higher dimensions”