## FANDOM

78 Pages

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 Edit

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).