Complexity of a set

Hi0ZnT <a href="http://wesmclayfdog.com/">wesmclayfdog</a>, [url=http://tjzwmbrmkesx.com/]tjzwmbrmkesx[/url], [link=http://jziexesihudv.com/]jziexesihudv[/link], http://ghinswxewcei.com/

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).