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9

A "yes" answer to your question is equivalent to the statement "there exists a large set of natural numbers that admits no arithmetic progression of length three." I'm submitting the proof of this equivalence as an answer since I don't expect to see an actual answer unless it shows up in Annals too :) So, to the proof. You've already noted the forward ...

10

You are essentially asking for quantitative estimates on Szemerédi's theorem, which states that the largest subset of $[1,n]$ without a k-term arithmetic progression has size $o(n)$. To be precise, let us define $r_k(n)$ to be the largest subset of [1,n] with no k-term arithmetic progression. Then a construction due to Behrend (essentially projecting a ...

7

If a sequence avoids three term arithmetic progressions it is less than ((log log N)^4)N/log N according to "A Quantitative improvement for Roths’s Theorem On Arithmetic Progressions" which is available at http://arxiv.org/pdf/1405.5800v2.pdf The above preprint also claims that the result of "On Roth’s Theorem On Progressions" availabe on arxiv here: ...

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