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: $\bigwedge _{\alpha <\kappa }\phi _{\alpha }(x;b_{\alpha ,\eta (\alpha )})\wedge \bigwedge _{\alpha <\kappa ,i\neq \eta (\alpha )}\neg \phi _{\alpha }(x;b_{\alpha ,i}).$ 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 $\eta :\kappa \to \omega$ , $\Sigma (x)\wedge \bigwedge _{\alpha <\kappa }\phi _{\alpha }(x;b_{\alpha ,\eta (\alpha )})\wedge \bigwedge _{\alpha <\kappa ,i\neq \eta (\alpha )}\neg \phi _{\alpha }(x;b_{\alpha ,i})$ is consistent.

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