EDP17 — are we nearly there?

Apologies for the attention-seeking title, but that really is the purpose of this post. I want to draw attention to some ideas that are buried in more comments that most people are likely to want to read, because I think there is a chance that all that stands between where we are now and a solution to EDP is a few soluble technical problems. It goes without saying that that chance is distinctly less than 100%, but I think it is high enough for it to be worth my going to some trouble to lay out as precisely as I can what the current approach is and what remains to be done. I’ll try to write it in such a way that it explains what is going on even to somebody who has not read any of the posts or comments so far. The exception to that is that I shall not repeat very basic things such as what the Erdős discrepancy problem is, what a homogeneous arithmetic progression is, etc. For that kind of information, I refer the reader to the front page of the Polymath5 wiki.

Polymath news

If you haven’t already spotted this, you might like to know that Scott Aaronson has just posed a very interesting unsolved combinatorial problem and invited Polymath-style thoughts on it. I’ll give a quick and I hope appetite-whetting account of the problem here, but you could skip it if you want and jump straight to his posts on his blog and at Mathoverflow.

The problem is completely elementary, but it has consequences of great interest to theoretical computer scientists. As Scott says, it also seems to be at a realistic level of difficulty and highly suitable for a Polymath approach. (The first assertion is a coded way of saying that it isn’t a notorious “impossible” complexity assertion in disguise.) It asks this. Suppose you colour the points of with two colours in such a way that the origin is red and each of the coordinate axes contains at least one blue point. Do there exist constants , independent of , such that there must be a point that has at least neighbours that are coloured differently from ? At the moment, it seems that there isn’t even a proof that for sufficiently large there must be a point with 100 neighbours of the other colour. There is a non-trivial but not hard example that shows that cannot be more than . (Scott presents this in his Mathoverflow post.) A very brief remark is that if you just assume that not all points are coloured the same way, then the assertion is false: a counterexample is obtained if you colour a point red if its first coordinate is positive and blue otherwise.

EDP16 — from AP-discrepancy to HAP-discrepancy?

In this post I want to elaborate on a strategy for proving EDP that I discussed in EDP15. Briefly, the idea is to take a representation-of-identity proof of Roth’s AP-discrepancy result and modify it so that it becomes a representation-of-diagonal proof of unbounded HAP-discrepancy.

The first step of this programme was an obvious one: obtain a clean and fully detailed proof in the APs case. That has now been completed, and a write-up can be found here. For the benefit of anyone who is interested in thinking about the next stage but doesn’t feel like reading a piece of formal mathematics, let me give a sketch of the argument here. That way, this post will be self-contained. Once I’ve given the sketch, I’ll say what I can about where we might go from here. It is just possible that we are in sight of the finishing line, but that is unlikely as it would depend on various guesses being correct, various potential technical problems not being actual, various calculations giving strong enough bounds, and so on. Thus, the final part of this post will be somewhat speculative, but I will make it as precise as I can in the hope that it will give rise to a fruitful line of enquiry.