## FANDOM

78 Pages

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.