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If you were to 'color' these like a four color map problem, are they all trivially colorable with just two colors? It looks that way from the pictures, but I la
by showerst 14y ago
If you were to 'color' these like a four color map problem, are they all trivially colorable with just two colors? It looks that way from the pictures, but I lack the math intuition to prove it.
- shasta 14y agoYes, the intersection of a number of curves defining regions is color-able with two colors: any point covered by an odd number of regions is black and any point covered by an even number of regions is white (i.e. a checkerboard pattern). Works as long as the boundary intersections are transverse. The proof is simply that if you cross a boundary, you must be decreasing or increasing the region count by one, changing the parity.
- jgeralnik 14y agoEach color is touched by at most one color on top and one color on bottom. Looking at the original 7-Venn we get Red->Orange->Yellow->Green->Blue->Purple. Change the odd colors to white and the even colors to black and you're done.