TITLE: Prime-like numbers that avoid Green-Tao?
QUESTION [7 upvotes]: I would like to understand the conditions that support
the Green-Tao Theorem, which established that
the primes contain arbitrarily long arithmetic progressions.
I am wondering:

Q. Is it difficult to define an infinite
  set $S$ of natural numbers that mimics the 
  $\sim n / \log n$ distribution of
  the primes (and perhaps other prime properties), but yet fails to contain arbitrarily long arithmetic progressions?

For example, $S$ might be the
lucky numbers
(OEIS A000959),
or
Hawkins' "random primes."
Do either of these avoid arbitrarily long arithmetic progressions?
My intuition is that it is easier to avoid such progressions
than it is to establish their existence.

MSE question unanswered:
Do lucky numbers contain arbitrarily long arithmetic progressions?

REPLY [11 votes]: This is likely impossible. 
Indeed the largest sets known to be free of arbitrarily long arithmetic progressions asymptotically satisfy  $| [ A \cap [1, n] | \lesssim_{k} n / \log^{k} n$ for all $k>1$, and it is widely believed that these examples are near maximal.
The Erdos-Turan conjecture mentioned in the comments is almost equivalent to the claim that any set of relative density greater than $n/ \log n$ contains arbitrary long arithmetic progressions. As remarked above, this is likely true for even sparser sets.