Fix some theory . Let be a cardinal. An inp pattern of depth is a collection of formulas and constants for and and integers such that for every , the set of formulas is -inconsistent, but for every function , the collection is consistent.

More generally, if is a partial type, an inp pattern of depth in is an inp pattern of depth such that for every , is consistent.

Shelah defines of a theory to be the supremum of the depths of possible inp-patterns. Hans Adler (right?) defines the burden of a partial type to be the supremum of the depths of the inp patterns in . A theory is said to be strong if there are no inp patterns of depth . A theory is if and only if .

Artem Chernikov (right?) proved that burden is submultiplicative in the following sense: if and , then . It is conjectured that burden is subadditive (), but this is unknown.

Given an inp pattern of depth , one can always find an inp pattern of the same depth, using the same formulas and same 's, such that the rows are mutually indiscernible. Given mutual indiscernibility, the -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 to be the supremum of the for which there exists mutually indiscernible sequences for and formulas for such that for each , is inconsistent, and 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 can be converted to a mutually indiscernible inp pattern of the same depth, as follows. If the original ict pattern is , then we take as our inp pattern the array of formulas whose entry in the th row and th column is . The "no alternation" characterization of NIP implies that each row is inconsistent. The ict condition ensures that we can find an satisfying and for every , showing that the first column is consistent.

Consequently, if NIP holds (equivalently, ), then , 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). ::: ::: :::