The Hadwiger-Nelson problem is that of determining the chromatic number of the plane (\(CNP\)), defined as the minimum number of colours that can be assigned to the points of the plane so as to prevent any two points unit distance apart from being the same colour. It was first posed in 1950 and the bounds \(4 \leq CNP \leq 7\) were rapidly demonstrated, but no further progress has since been made. In a recent preprint, I have now excluded the case \(4 \leq CNP \leq 7\) by identifying a family of non-\(4\)-colourable finite “unit-distance” graphs, i.e. graphs that can be embedded in the plane with all edges being straight lines of length \(1\). However, the smallest such graph that I have so far discovered has \(1567\) [EDIT: \(1581\) after correction] vertices, and its lack of a \(4\)-colouring requires checking for the nonexistence of a particular category of \(4\)-colourings of subgraphs of it that have \(388\) [EDIT: \(395\) after correction] and \(397\) vertices, which obviously requires a computer search.
I’m therefore wondering whether a search for “simpler” examples might work as a Polymath project. An example might be defined as simpler if it has fewer vertices, or if it has a smaller largest subgraph whose \(4\)-colourability must be checked directly, etc. I feel that a number of features make this nice for Polymath:
- being graph theory, it’s nicely accessible/seductive to non-specialists
- it entails a rich interaction between theory and computation
- simpler graphs may lead to insights into what properties such graphs will always/usually have, which might inspire strategies for seeking 6-chromatic examples, improved bounds to the analogous problem in higher dimensions, etc.
I welcome comments!
Aubrey de Grey
[Original text from Aubrey de Grey]