Sharp logarithmic-logarithmic factors in higher dimensions

Establish that, for every fixed dimension d≥3, the upper bound on the chromatic number and the lower bound on the independence number of random d-dimensional box-Delaunay graphs and Hasse diagrams have the correct log log n factor, namely that the relevant exponent is 1 up to constant factors.

Background

For uniformly random point sets in [0,1]^d with fixed d≥3, the paper proves upper and lower bounds whose principal factor is (log n)^{d−1}, but whose log log n exponents differ substantially. The authors conjecture that the upper chromatic-number bound and the lower independence-number bound can both be sharpened so that the exponent of log log n is exactly 1.

The stated problem concerns closing the quantitative gap left by the higher-dimensional random-point results, rather than proving a new qualitative property of the graphs. The authors explicitly note that they were unable to establish this sharper form.