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Topic: cotpi 69 - Black and white plane
Replies: 26   Last Post: Oct 9, 2013 10:23 AM

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 quasi Posts: 12,067 Registered: 7/15/05
Re: cotpi 69 - Black and white plane
Posted: Oct 2, 2013 10:32 PM

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.")

Nice result.

Here's an immediate corollary ...

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))?

quasi

Date Subject Author
10/2/13 cotpi
10/2/13 Pubkeybreaker
10/2/13 quasi
10/2/13 Mike Terry
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 quasi
10/6/13 quasi
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 quasi
10/6/13 David Bernier
10/9/13 Phil Carmody
10/9/13 David Bernier
10/2/13 Eric Lafontaine
10/2/13 Michael F. Stemper
10/2/13 quasi
10/2/13 quasi
10/2/13 Haran Pilpel
10/2/13 quasi
10/2/13 quasi
10/2/13 Ted Schuerzinger