Comments on: Proposal: Boshernitzan’s problem

By: Kevin O'Bryant

Kevin O'Bryant — Mon, 17 Aug 2009 06:06:43 +0000

Oops, this has been solved, too:

MR1881964 (2003b:05153)
Banakh, T. O.(UKR-LVV-MM); Kmit, I. Ya.(UKR-LST-NMP); Verbitsky, O. V.(UKR-LVV-MM)
On asymmetric colorings of integer grids. (English summary)
Ars Combin. 62 (2002), 257–271.

By: Kevin O'Bryant

Kevin O'Bryant — Sun, 09 Aug 2009 08:28:15 +0000

Another question of this type: Let be lattice points in dimensions, and suppose that is a finite set. Must the set contain arbitrarily large symmetric subsets?

The set is symmetric if there is a (not necessarily in ) such that . For example, arithmetic progressions are symmetric.

By: Kevin V.

Kevin V. — Tue, 04 Aug 2009 07:27:07 +0000

In all of the negative results given by Dekking, the counterexamples have step sizes that are not only bounded, but are actually all equal to one. However, in the case (d,k)=(2,3), one can verify by hand that if a counterexample exists, it must have step sizes greater than one.

For the two remaining cases, I wrote a little algorithm to find long paths which avoid AP’s, assuming we only allow a step size of one. It found paths of length > 80 for both (d,k)=(2,4) and (d,k)=(3,3) in a matter of minutes. This seems to suggest that it is possible to construct a counterexample in these cases. Unfortunately, the constructed paths don’t appear to follow any obvious pattern.

By: kristalcantwell

kristalcantwell — Mon, 03 Aug 2009 19:15:25 +0000

“It is also worth trying to find a counterexample if C, d, k are large enough. Note that the continuous analogue of the problem is false: a convex curve in the plane, such as the parabola \{ (x,x^2): x \in {\Bbb R}\}, contains no arithmetic progressions, but is rectifiable. However it is not obvious how to discretise this example.”

If the problem is changed so integers are replaced by real numbers then it is still false as there is only a finite number of points forming lines and an uncountable number of points that are real numbers distance C from the original point.

By: Terence Tao

Terence Tao — Mon, 03 Aug 2009 13:12:57 +0000

I’ve set up a rudimentary wiki page for this problem at

http://michaelnielsen.org/polymath1/index.php?title=Boshernitzan’s_problem

It also incorporates some information sent to me by email by Michael Boshneritzan. As always, further contributions are welcome (currently, for instance, Croot’s problems are not on the wiki).

By: gowers

gowers — Wed, 29 Jul 2009 16:16:18 +0000

It was the third one that I was referring to. I like the way it seems to invite a curious mixture of topological arguments with more conventional density ones — indeed, so curious that it isn’t obvious at all how a proof could go if the answer was positive.

By: kristalcantwell

kristalcantwell — Wed, 29 Jul 2009 16:02:15 +0000

If the following is true:

Dekking shows that there is an inﬁnite sequence of plane lattice points where each gap is (0, 1) or (1, 0) such that no 5 points are in AP.

Don’t we have a complete solution to the problem as posted?

If dimension is one we are done.
If we have two linearly independent possible steps we can use them in place of the gaps (0,1) and (1,0) in Dekkings proof and limit arithmetic progressions to length 5 which solves the problem as stated since we are trying to block arbitrarily long arithmetic progressions and we can in fact block all those of length six.

By: Anonymous

Anonymous — Wed, 29 Jul 2009 15:26:58 +0000

Can anyone post the Dekking paper or give a short summary of the construction? I couldn’t find the paper on the web and my (like most) library doesn’t have access to JCT volumes as far back as 1979.

By: Ernie Croot

Ernie Croot — Wed, 29 Jul 2009 14:38:58 +0000

Thanks! The third question in the note I feel is probably quite hard, but the second one can surely be worked out easily. I suppose one can think of the third problem as a type of “Turan-type analogue” to Michael B.’s problem.

On an unrelated matter, I am glad to hear about the paper of Dekking on paths without 5APs. I had come to that problem myself, and had asked two grad students here at Georgia Tech (my student Evan Borenstein and Michael Lacey’s student Bill McClain) about whether they could construct such paths (without 5APs), but we didn’t ever produce one, though we sort of knew how to do it.

By: gowers

gowers — Wed, 29 Jul 2009 14:15:52 +0000

If it doesn’t turn out to have an unexpectedly easy solution, then that’s a very nice question!