Haran Pilpel wrote: >quasi writes: > >> Does there exist a 2-coloring of Q^2 such that no two points >> of Q^2 which are 1 unit apart have the same color? > >Yes. This is a result of Woodall from 1973. ("Distances realized by sets covering the plane.")
Here's an immediate corollary ...
If a closed piecewise-linear path consists of n unit-length segments with rational endpoints, then n is even.
Some followup questions ...
For each positive integer n, let
D_n be the disk in Q^2, centered at the origin, with radius n.
E_n be the set of possible terminal points of piecewise-linear paths, which start at the origin and consist of k unit-length segments, where 0 <= k <= n.
Question (1): Is it true that, for sufficiently large n, E_n = D_n?
If the answer to question (1) is no, then
Question (2): Is it true that, for sufficiently large n, D_1 subset E_n?
If the answer to question (2) is no, then
Question (3): Is it true that, for sufficiently large n, (D_1 /\ E_n) = (D_1 /\ E_(n+1))?