Complexity of a set
Sets of complexity 1 in [math]\displaystyle{ [3]^n }[/math]
Definition
Let [math]\displaystyle{ \mathcal{U}, \mathcal{V} }[/math] and [math]\displaystyle{ \mathcal{W} }[/math] be collections of subsets of [math]\displaystyle{ [n]. }[/math] Then we can define a subset [math]\displaystyle{ \mathcal{A} }[/math] of [math]\displaystyle{ [3]^n }[/math] by taking the set of all sequences x such that the 1-set of x (meaning the set of coordinates i where [math]\displaystyle{ x_i=1 }[/math]) belongs to [math]\displaystyle{ \mathcal{U}, }[/math] the 2-set of x belongs to [math]\displaystyle{ \mathcal{V} }[/math] and the 3-set of x belongs to [math]\displaystyle{ \mathcal{W}. }[/math] If [math]\displaystyle{ \mathcal{A} }[/math] can be defined in this way, then we say that it has complexity 1. DHJ(1,3) is the special case of DHJ(3) that asserts that a dense set of complexity 1 contains a combinatorial line.
Motivation
Sets of complexity 1 are closely analogous to sets that arise in the theory of 3-uniform hypergraphs. One way of constructing a 3-uniform hypergraph H is to start with a graph G and let H be the set of all triangles in G (or more formally the set of all triples xyz such that xy, yz and xz are edges of G). These sets form a complete set of obstructions to uniformity for 3-uniform hypergraphs, so there is reason to expect that sets of complexity 1 will be of importance for DHJ(3).
Special sets of complexity 1
A more restricted notion of a set of complexity 1 is obtained if one assumes that [math]\displaystyle{ \mathcal{W} }[/math] consists of all subsets of [math]\displaystyle{ [n]. }[/math] We say that [math]\displaystyle{ \mathcal{A} }[/math] is a special set of complexity 1 if there exist [math]\displaystyle{ \mathcal{U} }[/math] and [math]\displaystyle{ \mathcal{V} }[/math] such that [math]\displaystyle{ \mathcal{A} }[/math] is the set of all [math]\displaystyle{ x\in[3]^n }[/math] such that the 1-set of x belongs to [math]\displaystyle{ \mathcal{U} }[/math] and the 2-set of x belongs to [math]\displaystyle{ \mathcal{V}. }[/math] Special sets of complexity 1 appear as local obstructions to uniformity in DHJ(3). (See this article for details.)
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).
