1Department of Computer Science and Software Engineering, Concordia University, Montréal, Canada. 2Department of Mathematics, University of Idaho - Moscow, Idaho, USA.
The electronic journal of combinatorics, Tome 20 (2013) no. 4
A planar point-set $X$ in Euclidean plane is called a $k$-distance set if there are exactly $k$ different distances among the points in $X$. The function $g(k)$ denotes the maximum number of points in the Euclidean plane that is a $k$-distance set. In 1996, Erdős and Fishburn conjectured that for $k\geq 7$, every $g(k)$-point subset of the plane that determines $k$ different distances is similar to a subset of the triangular lattice. We believe that if $g(k)$ is an increasing function of $k$, then the conjecture is false. We present data that supports our claim and a method of construction that unifies known optimal point configurations for $k\geq 3$.
1
Department of Computer Science and Software Engineering,
Concordia University, Montréal, Canada.
2
Department of Mathematics,
University of Idaho - Moscow, Idaho, USA.
@article{10_37236_3263,
author = {Tanbir Ahmed and Hunter Snevily},
title = {Sparse distance sets in the triangular lattice},
journal = {The electronic journal of combinatorics},
year = {2013},
volume = {20},
number = {4},
doi = {10.37236/3263},
zbl = {1295.52011},
url = {http://geodesic.mathdoc.fr/articles/10.37236/3263/}
}
TY - JOUR
AU - Tanbir Ahmed
AU - Hunter Snevily
TI - Sparse distance sets in the triangular lattice
JO - The electronic journal of combinatorics
PY - 2013
VL - 20
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/3263/
DO - 10.37236/3263
ID - 10_37236_3263
ER -
%0 Journal Article
%A Tanbir Ahmed
%A Hunter Snevily
%T Sparse distance sets in the triangular lattice
%J The electronic journal of combinatorics
%D 2013
%V 20
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/3263/
%R 10.37236/3263
%F 10_37236_3263
Tanbir Ahmed; Hunter Snevily. Sparse distance sets in the triangular lattice. The electronic journal of combinatorics, Tome 20 (2013) no. 4. doi: 10.37236/3263
