Fix some theory *T*. Let *κ* be a cardinal. An **inp pattern** of depth *κ* is a collection of formulas ⟨*ϕ*_{α}(*x*; *y*)⟩_{α < κ} and constants *b*_{α, i} for *α* < *κ* and *i* < *ω* and integers *k*_{α} < *ω* such that for every *α* < *κ*, the set of formulas {*ϕ*(*x*; *b*_{α, i}) : *i* < *ω*} is *k*_{α}-inconsistent, but for every function *η* : *κ* → *ω*, the collection {*ϕ*(*x*; *b*_{α, η(α)}) : *α* < *κ*} is consistent.

More generally, if *Σ*(*x*) is a partial type, an inp pattern of depth *κ* in *Σ*(*x*) is an inp pattern of depth *κ* such that for every *η* : *κ* → *ω*, *Σ*(*x*) ∪ {*ϕ*(*x*; *b*_{α, η(α)}) : *α* < *κ*} is consistent.

Shelah defines *κ*_{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 *Σ*(*x*) to be the supremum of the depths of the inp patterns in *Σ*(*x*). A theory is said to be **strong** if there are no inp patterns of depth *ω*. A theory is *N**T**P*_{2} if and only if *κ*_{inp} < ∞.

Artem Chernikov (right?) proved that burden is submultiplicative in the following sense: if *b**d**n*(*b*/*C*) < *κ* and *b**d**n*(*a*/*b**C*) < *λ*, then *b**d**n*(*a**b*/*C*) < *κ* × *λ*. It is conjectured that burden is subadditive (*b**d**n*(*a**b*/*C*) ≤ *b**d**n*(*a*/*b**C*) + *b**d**n*(*b*/*C*)), but this is unknown.

Given an inp pattern of depth *κ*, one can always find an inp pattern of the same depth, using the same formulas and same *k*_{α}'s, such that the rows ⟨*b*_{α, i}⟩_{i < ω} are mutually indiscernible. Given mutual indiscernibility, the *k*_{α}-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 *Σ*(*x*) to be the supremum of the *κ* for which there exists *κ* mutually indiscernible sequences ⟨*b*_{α, i}⟩_{i} for *α* < *κ* and formulas *ϕ*_{α}(*x*; *y*) for *α* < *κ* such that for each *α*, {*ϕ*_{α}(*x*; *b*_{α, i}) : *i* < *ω*} is inconsistent, and *Σ*(*x*) ∪ {*ϕ*_{α}(*x*; *b*_{α, 0}) : *α* < *κ*} 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 *κ* can be converted to a mutually indiscernible inp pattern of the same depth, as follows. If the original ict pattern is {*ϕ*_{α}(*x*; *b*_{α, i}), then we take as our inp pattern the array of formulas whose entry in the *α*th row and *i*th column is *ϕ*_{α}(*x*; *b*_{α, 2i}) ∧ ¬*ϕ*_{α}(*x*; *b*_{α, 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 *ϕ*_{α}(*x*; *b*_{α, 0}) and ¬*ϕ*_{α}(*x*; *b*_{α, 1}) for every *α*, showing that the first column is consistent.

Consequently, if NIP holds (equivalently, *κ*_{ict} < ∞), then *κ*_{inp} = *κ*_{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). ::: ::: :::