There has been further recent activity on the Chromatic Number of the Plane problem, with an eleventh research thread being spawned. Philip Gibbs has been able to 6-colour a large disc (with diameter slightly greater than 4), and Aubrey de Gray has remarked that it can be enlarged slightly further still:
An infinite strip of width can similarly be 6-coloured in a relatively simple way.
What about the whole plane?
Interestingly, it has been shown that any tile-based 6-colouring of the plane is critical in the sense that the maximum diameter of any tile must be equal to the minimum separation between similarly-coloured tiles; there is no room for manoeuvre. Moreover, this means that it is insufficient to simply specify the colours of the tiles themselves; it is necessary to also colour the (measure-0) vertices and edges where they meet!
More updates as events warrant…