FANDOM


Fix some theory $ T $. Let $ \kappa $ be a cardinal. An ict pattern of depth $ \kappa $ is a collection of formulas $ \langle \phi_\alpha(x;y) \rangle_{\alpha < \kappa} $ and constants $ b_{\alpha,i} $ for $ \alpha < \kappa $ and $ i < \omega $, such that for every $ \eta : \kappa \to \omega $, the following collection of formulas is consistent: $ \bigwedge_{\alpha < \kappa} \phi_\alpha(x;b_{\alpha,\eta(\alpha)}) \wedge \bigwedge_{\alpha < \kappa, i \ne \eta(\alpha)} \neg \phi_\alpha(x;b_{\alpha,i}). $ So we have an array of formulas, with $ \kappa $ rows and $ \omega $ columns, each row being uniform, and for every vertical path through the array, there is an element which satisfies exactly those formulas along the path, and no others.

More generally, if $ \Sigma(x) $ is a partial type over some parameters, then an ict pattern of depth $ \kappa $ in $ \Sigma(x) $ is an array as above, such that for each $ \eta : \kappa \to \omega $, $ \Sigma(x) \wedge \bigwedge_{\alpha < \kappa} \phi_\alpha(x;b_{\alpha,\eta(\alpha)}) \wedge \bigwedge_{\alpha < \kappa, i \ne \eta(\alpha)} \neg \phi_\alpha(x;b_{\alpha,i}) $ is consistent.

Given an ict pattern, we can always extract another ict pattern using the same formulas, but with the $ \langle b_{\alpha,i} \rangle $ mutually indiscernible.

Shelah defines $ \kappa_{ict} $ of the theory $ T $ to be the supremum of the depths of ict patterns, or $ \infty $ if there exist ict patterns of unbounded depth. It turns out that $ \kappa_{ict} < \infty $ if and only if $ T $ is NIP.

A theory is said to be strongly dependent if there are no ict patterns of depth $ \aleph_0 $. The maximum depth of an ict pattern in a type $ \Sigma(x) $ is the dp-rank of $ \Sigma(x) $, or some variant thereof.