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 [here](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](http://arxiv.org/pdf/1405.5800v2.pdf)
is in error in claiming O(((log log N)^5 N/log N) and that it should be
O(((log log N)^6 N/log N).

So the upper bound even for three term arithmetic progressions is at most a factor of log log N to the fourth power better than N/log N if the preprint in the first paragraph is correct.