6-colourings of subsets of the plane

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…