INP pattern

Fix some theory $$T$$. Let $$\kappa$$ be a cardinal. An inp 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 $$i < \omega$$ and integers $$k_\alpha < \omega$$ such that for every $$\alpha < \kappa$$, the set of formulas $$\{\phi(x;b_{\alpha,i}) : i < \omega\}$$ is $$k_\alpha$$-inconsistent, but for every function $$\eta : \kappa \to \omega$$, the collection $$\{\phi(x;b_{\alpha,\eta(\alpha)}) : \alpha < \kappa\}$$ is consistent.

More generally, if $$\Sigma(x)$$ is a partial type, an inp pattern of depth $$\kappa$$ in $$\Sigma(x)$$ is an inp pattern of depth $$\kappa$$ such that for every $$\eta : \kappa \to \omega$$, $$\Sigma(x) \cup \{\phi(x;b_{\alpha,\eta(\alpha)}) : \alpha < \kappa\}$$ is consistent.

Shelah defines $$\kappa_{inp}$$ of a theory to be the supremum of the depths of possible inp-patterns. Hans Adler (right?) defines the burden of a partial type $$\Sigma(x)$$ to be the supremum of the depths of the inp patterns in $$\Sigma(x)$$. A theory is said to be strong if there are no inp patterns of depth $$\omega$$. A theory is $$NTP_2$$ if and only if $$\kappa_{inp} < \infty$$.

Artem Chernikov (right?) proved that burden is submultiplicative in the following sense: if $$bdn(b/C) < \kappa$$ and $$bdn(a/bC) < \lambda$$, then $$bdn(ab/C) < \kappa \times \lambda$$. It is conjectured that burden is subadditive ($$bdn(ab/C) \le bdn(a/bC) + bdn(b/C)$$), but this is unknown.

Given an inp pattern of depth $$\kappa$$, one can always find an inp pattern of the same depth, using the same formulas and same $$k_\alpha$$'s, such that the rows $$\langle b_{\alpha,i}\rangle_{i < \omega}$$ are mutually indiscernible. Given mutual indiscernibility, the $$k_\alpha$$-inconsistence can be rephrased as inconsistency. And the only vertical path one must check is the leftmost column. So one may also define the burden of $$\Sigma(x)$$ to be the supremum of the $$\kappa$$ for which there exists $$\kappa$$ mutually indiscernible sequences $$\langle b_{\alpha,i} \rangle_i$$ for $$\alpha < \kappa$$ and formulas $$\phi_\alpha(x;y)$$ for $$\alpha < \kappa$$ such that for each $$\alpha$$, $$\{\phi_\alpha(x;b_{\alpha,i}) : i < \omega\}$$ is inconsistent, and $$\Sigma(x) \cup \{\phi_\alpha(x;b_{\alpha,0}) : \alpha < \kappa\}$$ is consistent.

Relation to ict patterns
Any mutually indiscernible inp pattern is already a mutually indiscernible ict pattern. Under the hypothesis of NIP, a mutually indiscernible ict pattern of depth $$\kappa$$ can be converted to a mutually indiscernible inp pattern of the same depth, as follows. If the original ict pattern is $$\{\phi_\alpha(x;b_{\alpha,i})$$, then we take as our inp pattern the array of formulas whose entry in the $$\alpha$$th row and $$i$$th column is $$\phi_\alpha(x;b_{\alpha,2i}) \wedge \neg \phi_\alpha(x;b_{\alpha,2i+1})$$. The "no alternation" characterization of NIP implies that each row is inconsistent. The ict condition ensures that we can find an $$a$$ satisfying $$\phi_\alpha(x;b_{\alpha,0})$$ and $$\neg \phi_\alpha(x;b_{\alpha,1})$$ for every $$\alpha$$, showing that the first column is consistent.

Consequently, if NIP holds (equivalently, $$\kappa_{ict} < \infty$$), then $$\kappa_{inp} = \kappa_{ict}$$, and the burden of any type equals its dp-rank. Also, a theory is strongly dependent if and only if it is strong and NIP (dependent).