ICT pattern

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.