An upper bound for the minimum diameter of integral point sets

Abstract

For all n > d there exist n points in the Euclidean space E d such that not all points are in a hyperplane and all mutual distances are integral. It is proved that the minimum diameter of such integral point sets has an upper bound of 2 cl°gnl°gl°gn.

Maximal integral point sets over ℤ2

“…In addition to the construction of crabs, there is another useful construction of integral point sets of Z 2 with a large cardinality, see [10] for a similar construction over the ring Z −1+ √ −3 2 . Let p j ≡ 1 (mod 4) be distinct primes over N. We consider the ring Z[i], where every integer p j has a unique prime factorization p j = ω j · ω j , where c denotes the complex conjugate of c. We may write ω j = a j + b j i, with integers a j , b j .…”

Note on Integral Distances

“…For the minimum diameter of integral sets in general position, Harborth et al gave an upper bound [6]. They found integral point sets H of cardinality |H | = n on a circle whose diameter satisfies the inequality Diam(H ) < n c 1 log log(n) for a constant c 1 > 0.…”

“…When the points of an integral set are not necessarily in general position, the diameter can be as small as n − 1, if the points are on one line. In [6] Harborth et al determined the minimum diameter of non-collinear integral point sets of cardinality n for small values of n. The minimum diameters for n = 3, 4, 5, 6, 7, 8, 9 are 1, 4, 7, 8, 17, 21, 29, respectively. Now we show that the diameter is always at least linear in n. Proof.…”

The maximum number of odd integral distances between points in the plane

“…This maximum f(n) is determined in Theorem 1. The proof is based on a construction given in [3]. Moreover, we make use of Turfin's theorem from graph theory (see p. 30 of [5] For the third root of unity p = (-1 + q/3i)/2 with i 2 = -1, and for 09 = 3 + p let Ok = 7,.-Lk/3jP kw2[k/aj, and pk = ~r/kz/(3 9 7,.)…”

“…with 0 < k < 3m. This is the special case R = 7" = (w&)," of a construction given in [3] (see Fig. 1 for the case m = 2).…”