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.

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

]]>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.

]]>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.

]]>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).

]]>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.

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.

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