Complexity of a set
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Sets of complexity j in [math]\displaystyle{ [k]^n }[/math]
We can make a similar definition for sequences in [math]\displaystyle{ [k]^n }[/math], or equivalently ordered partitions [math]\displaystyle{ (U_1,\dots,U_k) }[/math] of [math]\displaystyle{ [n]. }[/math] Suppose that for every set [math]\displaystyle{ E }[/math] of size j there we have a collection [math]\displaystyle{ \mathcal{U}_E }[/math] of j-tuples [math]\displaystyle{ (U_i:i\in E) }[/math] of disjoint subsets of [math]\displaystyle{ [n] }[/math] indexed by [math]\displaystyle{ E. }[/math] Then we can define a set system [math]\displaystyle{ \mathcal{A} }[/math] to consist of all ordered partitions [math]\displaystyle{ (U_1,\dots,U_k) }[/math] such that for every [math]\displaystyle{ E\subset\{1,2,\dots,k\} }[/math] of size j the j-tuple of disjoint sets [math]\displaystyle{ (U_i:i\in E) }[/math] belongs to [math]\displaystyle{ \mathcal{U}_E. }[/math] If [math]\displaystyle{ \mathcal{A} }[/math] can be defined in that way then we say that it has complexity j.
DHJ(j,k) is the assertion that every subset of [math]\displaystyle{ [k]^n }[/math] of complexity j contains a combinatorial line. It is not hard to see that every subset of [math]\displaystyle{ [k]^n }[/math] has complexity at most [math]\displaystyle{ k-1, }[/math] so DHJ(k-1,k) is the same as DHJ(k).
