Centre d-cube at x in |Z^d , where 0 <= d <= |N ; Denote colour C by integer in {-d, ..., +d} Define C(X) = \sum_{0 <= i <= d} i x_i mod (2 d + 1) ; the sum remains finite since bounded by |x| max(i | X_i <> 0) . WFL On 9/29/20, Dan Asimov <dasimov@earthlink.net> wrote:
Let the coordinates of (infinite-dimensional) Hilbert space H be denoted as x_1, x_2, x_3, ..., x_n, ....
Note that the integer points of H correspond to arbitrary sequences of integers *only finitely many of which* are non-zero.
The cubic tiling of H has one cube for each integer point of H.
Definition: ----------- The cube at the integer point x is "adjacent" to the cube at the integer point y exactly if y differs from x in precisely one coordinate, by the amount ±1 in that coordinate, and has all the other coordinates equal to those of x.
Puzzle: ------- Prove that given infinitely many colors, indexed by the set of integers, the cubes of the cubic tiling of H can be colored so that each cube is adjacent to exactly one cube of each other color (and not adjacent to any cube of its own color).
—Dan
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