# infinite combinatorics – Set “crossing” all arithmetical integer sets

Yes. Enumerate all arithmetic sets $$A_1, A_2, ldots$$. Choose $$n_kin A_k$$ with $$|n_k|>2^k$$. The set $${n_1, n_2, ldots}$$ intersects any arithmetic set by construction, but it has zero density, thus does not contain a whole arithmetic set.

Actually you may construct such $$S$$ for each infinite sequence $$A_1, A_2, ldots$$ of infinite sets: on $$k$$-th step choose $$n_kin A_kcap S$$ and $$m_kin A_ksetminus S$$ so that $$n_1, m_1, n_2, m_2, ldots$$ remain distinct.