Chromatic Number of the Plane in Special Circumstances

As you know from Chaps. 4 and 6 , 3 years after Dmitry E. Raiskii, in 1973, Douglas R. Woodall published the paper [Woo1] on problems related to the chromatic number of the plane. In it, he provided his own proofs of Raiskii’s inequalities of problems 4.1 and 6.1. In the same paper, Woodall also formulated and attempted to prove a lower bound for the chromatic number of the plane for the special case of map-type coloring of the plane. This was the main result of [Woo1]. However, in 1979, the mathematician from the University of Aberdeen Stephen Phillip Townsend found an error in Woodall’s proof and constructed a counterexample, demonstrating that one essential idea of Woodall’s proof was false. Townsend had also found a proof of Woodall’s statement, which was very much more elaborate than Woodall’s unsuccessful attempt.