Closing the higher-dimensional chromatic and independence gaps

Close the gaps between the upper and lower bounds for the chromatic number and independence number of random-point Hasse diagrams and box-Delaunay graphs in fixed dimensions d≥3.

Background

The paper establishes matching principal powers of log n for the typical chromatic and independence numbers in dimensions d≥3, but the displayed bounds contain different powers of log log n. The authors identify closing these upper–lower gaps as an open direction immediately following their main higher-dimensional theorem.

This problem is a direct formulation of the unresolved quantitative discrepancy in the theorem and is closely related to the authors’ conjecture that the correct log log n exponent is 1.