This post is intended as a launch of Polymath9. I have no idea how the project will go, but I think it may be rather short lived, since the difficulties I am having at the moment look as though they could turn out to be serious ones that rule out any approach along the lines I have been thinking about. However, it is difficult to say that with any certainty, because the approach is fairly flexible, so even if the precise statements I have been trying to prove are false, it might be possible to come up with variants that are true. In a way I find that a good state of affairs, because it increases the chances of proving something interesting. Obviously it increases the chances of proving that PNP if one has more ways of attacking the problem. (I’m not claiming that it increases the probability to one that is not small — just that it increases it.) But it also increases the chances of what I would regard as a very nice consolation prize if, as expected, the approach does not work, namely a new barrier to proving that PNP. I don’t think it would be as fundamental a barrier as the three main barriers discovered so far, since it would not be showing that existing methods cannot work. Rather, it would be saying, “Here’s something we could try. Oh dear, it doesn’t work.” But as long as that something was reasonably general, I think its failure to work could be interesting enough to publish.
I’ve thought a little about what phrase to attach to the project (the equivalent of “density Hales-Jewett” or “Erdős discrepancy problem”). I don’t want to call it “P versus NP” because that is misleading: the project I have in mind is much more specific than that. It is to assess whether there is any possibility of proving complexity lower bounds by drawing inspiration from Martin’s proof of Borel determinacy. Only if the answer turned out to be yes, which for various reasons seems unlikely at the moment, would it be reasonable to think of this as a genuine attack on the P versus NP problem. So the phrase I’ve gone for is “discretized Borel determinacy”. That’s what DBD stands for above. It’s not a perfect description, but it will do.