Every point in 3Dspace is colored either red, green or blue. Let R (resp. G and B) be the set of distances between red (resp. green and blue) points. Prove that at least one of R, G, or B, consists of all the nonnegative real numbers.
NOTATION:
R* = { P in E^3  P is colored red }
G* = { P in E^3  P is colored green }
B* = { P in E^3  P is colored blue }
R = { d(P,Q)  P,Q in R* }
G = { d(P,Q)  P,Q in G* }
B = { d(P,Q)  P,Q in B* }
X << Y denotes X is a subset of Y
X ^ Y denotes the intersection of sets X and Y
X v Y denotes the union of sets Y and Y
X  Y denotes the set difference of sets X and Y
sphere(X,y) denotes the sphere with center X
and radius y
PROOF (Others can flesh out the details):
Assume r in [0,inf)  R,
g in [0,inf)  G, and
b in [0,inf)  B
WOLOG let 0 < b <= g <= r.
Pick a point L in R*. Then sphere(L,r) << G* v B*.
Case A: sphere(L,r) << G*
Pick a point M in sphere(L,r). Then
sphere(M,g) ^ sphere(L,r) << G*
This is a contradiction since it is a
space circle containing colorless points.
Case B: sphere(L,r) << B*
Pick a point M in sphere(L,r). Then
sphere(M,b) ^ sphere(L,r) << B*
This is a contradiction since it is a
space circle containing colorless points.
Case C: (sphere(L,r) ^ B*) and (sphere(L,r) ^ G*)
are both nonempty
Pick a point M in sphere(L,r) ^ G*. Then
sphere(M,g) ^ sphere(L,r) << B*. Pick a
point N in sphere(M,g) ^ sphere(L,r). Then
sphere(N,b) ^ sphere(M,g) ^ sphere(L,r) << B*
This is a contradiction since it is a set
containing two colorless points.
Therefore, at least one of the sets R, G, or B must
equal [0,inf).
Oops  Back slashes don't work to well.
To clean up the details:
Case 1: 0 < b <= g <= r
This is the case discribed above.
Case 2: 0 <= g <= r
This case is described in Case A above.
Case 3: 0 <= 0 <= r
No points colored green or blue. Clearly,
R = [0,inf)
Case 4: 0 <= 0 <= 0
All points are colorless. Contradicts problem statement.
Edited on August 25, 2006, 9:41 pm
Edited on August 25, 2006, 9:49 pm
Edited on August 27, 2006, 11:54 am

Posted by Bractals
on 20060825 17:23:07 