We show that the number of unit distances determined by n points in ℝ3 is O(n3/2), slightly improving the bound of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl [5], established in 1990. The new… Expand

It is proved that for d=3, k-reptile simplices (tetrahedra) exist only for k=m3, which partially confirms a conjecture of Hertel, asserting that the only k- reptile tetrahedr are the Hill tetrahedral simplices.Expand

These results imply a tower function of Ω(n) of height d as a lower bound, matching an upper bound by Suk up to the constant in front of n, and provide a natural geometric Ramsey-type theorem with a large Ramsey function.Expand

This work constructed the first examples of semialgebraic predicates with the Ramsey function bounded from below by a tower function of arbitrary height: for every $k\ge 4$ they exhibit a $k-ary $\Phi$ in dimension $2^{k-4}$ with $R_\ Phi$ bounded below byA tower of h...Expand

An upper bound valid for an arbitrary $\mathcal{F}$ is provided: it is shown that every system of n sets with m non-empty fields in the Venn diagram admits an inclusion–exclusion formula with mO(log2n) terms and with ±1 coefficients, and that such a formula can be computed in mO (log 2n) expected time.Expand

A d-dimensional simplex S is called a k-reptile (or a k reptile simplex) if it can be tiled without overlaps by k simplices with disjoint interiors that are all mutually congruent and similar to S.… Expand

Let F = (F 1, F 2, …, F n) be a family of n sets on a ground set S, such as a family of balls in R d. For every finite measure μ on S, such that the sets of F are measurable, the classical… Expand