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

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

Shelah defines {\displaystyle \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 {\displaystyle \Sigma (x)} to be the supremum of the depths of the inp patterns in {\displaystyle \Sigma (x)}. A theory is said to be strong if there are no inp patterns of depth {\displaystyle \omega }. A theory is {\displaystyle NTP_{2}} if and only if {\displaystyle \kappa _{inp}<\infty }.

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

Given an inp pattern of depth {\displaystyle \kappa }, one can always find an inp pattern of the same depth, using the same formulas and same {\displaystyle k_{\alpha }}'s, such that the rows {\displaystyle \langle b_{\alpha ,i}\rangle _{i<\omega }} are mutually indiscernible. Given mutual indiscernibility, the {\displaystyle 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 {\displaystyle \Sigma (x)} to be the supremum of the {\displaystyle \kappa } for which there exists {\displaystyle \kappa } mutually indiscernible sequences {\displaystyle \langle b_{\alpha ,i}\rangle _{i}} for {\displaystyle \alpha <\kappa } and formulas {\displaystyle \phi _{\alpha }(x;y)} for {\displaystyle \alpha <\kappa } such that for each {\displaystyle \alpha }, {\displaystyle \{\phi _{\alpha }(x;b_{\alpha ,i}):i<\omega \}} is inconsistent, and {\displaystyle \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 {\displaystyle \kappa } can be converted to a mutually indiscernible inp pattern of the same depth, as follows. If the original ict pattern is {\displaystyle \{\phi _{\alpha }(x;b_{\alpha ,i})}, then we take as our inp pattern the array of formulas whose entry in the {\displaystyle \alpha }th row and {\displaystyle i}th column is {\displaystyle \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 {\displaystyle a} satisfying {\displaystyle \phi _{\alpha }(x;b_{\alpha ,0})} and {\displaystyle \neg \phi _{\alpha }(x;b_{\alpha ,1})} for every {\displaystyle \alpha }, showing that the first column is consistent.

Consequently, if NIP holds (equivalently, {\displaystyle \kappa _{ict}<\infty }), then {\displaystyle \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). ::: ::: :::