Fix some theory {\displaystyle T}. Let {\displaystyle \kappa } be a cardinal. An ict pattern of depth {\displaystyle \kappa } is a collection of formulas {\displaystyle \langle \phi _{\alpha }(x;y)\rangle _{\alpha <\kappa }} and constants {\displaystyle b_{\alpha ,i}} for {\displaystyle \alpha <\kappa } and {\displaystyle i<\omega }, such that for every {\displaystyle \eta :\kappa \to \omega }, the following collection of formulas is consistent: {\displaystyle \bigwedge _{\alpha <\kappa }\phi _{\alpha }(x;b_{\alpha ,\eta (\alpha )})\wedge \bigwedge _{\alpha <\kappa ,i\neq \eta (\alpha )}\neg \phi _{\alpha }(x;b_{\alpha ,i}).} So we have an array of formulas, with {\displaystyle \kappa } rows and {\displaystyle \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 {\displaystyle \Sigma (x)} is a partial type over some parameters, then an ict pattern of depth {\displaystyle \kappa } in {\displaystyle \Sigma (x)} is an array as above, such that for each {\displaystyle \eta :\kappa \to \omega }, {\displaystyle \Sigma (x)\wedge \bigwedge _{\alpha <\kappa }\phi _{\alpha }(x;b_{\alpha ,\eta (\alpha )})\wedge \bigwedge _{\alpha <\kappa ,i\neq \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 {\displaystyle \langle b_{\alpha ,i}\rangle } mutually indiscernible.

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

A theory is said to be strongly dependent if there are no ict patterns of depth {\displaystyle \aleph _{0}}. The maximum depth of an ict pattern in a type {\displaystyle \Sigma (x)} is the dp-rank of {\displaystyle \Sigma (x)}, or some variant thereof. ::: ::: :::