Some Reflections on the Four Colour Conjecture. A MISTAKE HAS BEEN FOUND IN THE ARGUMENT OF BLOG 44.

This will be reviewed in Blog 46. The problem is that the removal of Q and R from the network is possible given the 4-colouring established by the initial P, A_x, C_x arguments, but this 4-colouring won’t necessarily (likely) remain in place if a same-coloured semi-adjacent pair of nodes is going to be imposed elsewhere in the network. 

In other words, the new 4-colouring needed to accommodate this imposed semi-adjacent pair won’t necessarily result in the 3-colouring of the pentagons which were originally circuits of Q and R.

This is similar the kind of hiccups discussed in depth by Imre Lakatos in his famous Proofs and Reputations. A new line of reasoning opens up new configurations, but there are many alternative ways to proceed. Clearly the argument of Blog 44 won’t do, but there are many variations which need to be explored.

The analysis of the characteristics of these plane networks in the two previous blogs provides extra insights which may be needed. 

That the 4-colouring-plus argument by induction is worth pursuing is evident, because induction is ideally suited to the problem, and the simplest application of the induction principle doesn’t work. 

Inquiry will continue for another month in the hope that something positive can be salvaged from the wreck.

CHRISTOPHER ORMELL 7^th February 2024
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