  We investigate higher dimensional chain conditions, where the largeness
notion is given by Fubini products of a given ideal. From strong saturation
properties of an ideal, we derive abstractly versions of higher dimensional
$\Delta$-system lemma, which imply many posets, including any finite support
iteration of $\sigma$-centered posets and measure algebras, satisfy the higher
dimensional chain conditions. We then show that if a poset satisfies a
strengthening of the $\sigma$-finite chain condition by Horn and Tarski, then
it satisfies higher dimensional chain conditions. As an application, we derive
Ramsey-theoretic consequences, namely various partition hypotheses as studied
by Bannister, Bergfalk, Moore and Todorcevic, from the existence of ideals
satisfying strong chain conditions.
