
Possible progress on the Hadwiger-Nelson problem (chromatic number of the plane) - ginnungagap
https://plus.google.com/+TerenceTao27/posts/QBxTFAsDeBp
======
ginnungagap
Consider a graph on R^2 where two points are joined by an edge iff they have
distance 1, in the 50s people wondered what the chromatic number of this graph
is and established the trivial lower bound of 4 and upper bound of 7.

By a classic result of De Bruijn and Erdos this problem is equivalent to
finding the maximum chromatic number of a finite unit distance graph in the
plane, but no further progress was made in decades.

In a preprint published yesterday De Grey claims to have constructed a 1567
vertexes graph which can't be 4-coloured, as can be seen in the comment
section of his polymath proposal this was raised to 1585, the
non-4-colorability of this new graph seems to have been verified independently
with SAT solvers by a couple of people, so that's promising.

As usual that's huge if true, the result looks promising, but time will tell

