Beta-Skeleton Dilation

I'm writing this up postmortem in the sense that we have already threw in the towel and moved on, so please excuse the brevity.

For beta in the interval (0,1], the best lower bound on the dilation (== spanning ratio) of the beta-skeleton is given in

Beta-skeletons have unbounded dilation.
D. Eppstein.
Tech. Rep. 96-15, ICS, UCI, 1996.

And the best upper bound is given in
On the spanning ratio of Gabriel graphs and beta-skeletons, P. Bose, L. Devroye, W. Evans, and D. Kirkpatrick, Proceedings of the Fifth Latin American Symposium on Theoretical Informatics (LATIN), LNCS 2286, p. 479-493, April 2002.
(pdf)
Though both these bounds are respectively O(n^c) and O(n^d) where c and d are functions of beta in the range (0,1], there is a significant difference between them. The problem was to try to tighten either bound. See the cited papers for more information.