Is it possible to explicitly construct a set of integers $S$ which contains a positive proportion of the positive integers and every integer in $S$ is not divisible by any prime $p$ in the set of twin primes $\{3,5,7,11,13,17,19,29,31,\ldots\}$.
Clarification
Firstly, by twin prime I mean any prime $p$ where either $p+2$ or $p-2$ is prime.
It is known that approximately $83.83\%$ of all positive integers are divisible by a twin prime (see here). Therefore, approximately $16.17\%$ of all positive integers avoid being divisible by a twin prime.
I would like to see the explicit construction of a set $S$ whose elements avoid divisibility by twin primes. Moreover, I would like this set to contain a positive proportion of all positive integers, that is:
$$\sum_{\substack{n \leq x \\ n \in S}} 1 > c x$$
for some $c>0$.
Clearly, one could let $S$ equal the set of numbers not divisible by twin primes, but it would be interesting to see something more explicit such as an arithmetic progression (though I believe this to be impossible).