- The chromatic number of 2-space
- The chromatic number of 3-space
- The polychromatic number of 3-space
We say that we have a chromatic colouring of n-space if we assign each point of n-space to a set and there is some distance d (an excluded distance) so that no two points in the same set are distance d apart. The minimum number of sets needed is known as the chromatic number of n-space.
At the moment it is known only that the chromatic number of two space lies between 4 and 7 (inclusive) and that the chromatic number of three space lies between 5 and 15 (inclusive).
A polychromatic colouring of n-space is similar to a chromatic colouring of n-space but we allow each set to have an excluded distance that is different from another set (rather than a common one d).
There are polychromatic colourings of 2-space with 6 sets bettering the most efficient known chromatic colouring with 7 sets. It is not known whether 3-space admits a polychromatic colouring with less than 15 sets (the number of sets in the most efficient chromatic colouring of 3-space uses 15 sets).
Past Postgraduate Supervision
||"Irreducible representations of some classes of quantum laurent polynomials"
Past MSc Students
||"Quantizing properties of the 12-16 partition of 3-space"