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| ==Sets of complexity 1 in <math>[3]^n</math>==
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| ===Definition===
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| Let <math>\mathcal{U}, \mathcal{V}</math> and <math>\mathcal{W}</math> be collections of subsets of <math>[n].</math> Then we can define a subset <math>\mathcal{A}</math> of <math>[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>x_i=1</math>) belongs to <math>\mathcal{U},</math> the 2-set of x belongs to <math>\mathcal{V}</math> and the 3-set of x belongs to <math>\mathcal{W}.</math> If <math>\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]].
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| ===Motivation===
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| Sets of complexity 1 are closely analogous to sets that arise in the theory of [http://en.wikipedia.org/wiki/Hypergraph 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).
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| ===Special sets of complexity 1===
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| A more restricted notion of a set of complexity 1 is obtained if one assumes that <math>\mathcal{W}</math> consists of all subsets of <math>[n].</math> We say that <math>\mathcal{A}</math> is a ''special set of complexity 1'' if there exist <math>\mathcal{U}</math> and <math>\mathcal{V}</math> such that <math>\mathcal{A}</math> is the set of all <math>x\in[3]^n</math> such that the 1-set of x belongs to <math>\mathcal{U}</math> and the 2-set of x belongs to <math>\mathcal{V}.</math> Special sets of complexity 1 appear as local obstructions to uniformity in DHJ(3). (See [[Line-free sets correlate locally with complexity-1 sets|this article]] for details.)
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| ==Sets of complexity j in <math>[k]^n</math>==
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| We can make a similar definition for sequences in <math>[k]^n</math>, or equivalently ordered partitions <math>(U_1,\dots,U_k)</math> of <math>[n].</math>
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| Suppose that for every set <math>E</math> of size j there we have a collection <math>\mathcal{U}_E</math> of j-tuples <math>(U_i:i\in E)</math> of disjoint subsets of <math>[n]</math> indexed by <math>E.</math> Then we can define a set system <math>\mathcal{A}</math> to consist of all ordered partitions <math>(U_1,\dots,U_k)</math> such that for every <math>E\subset\{1,2,\dots,k\}</math> of size j the j-tuple of disjoint sets <math>(U_i:i\in E)</math> belongs to <math>\mathcal{U}_E.</math> If <math>\mathcal{A}</math> can be defined in that way then we say that it has ''complexity j''.
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| DHJ(j,k) is the assertion that every subset of <math>[k]^n</math> of complexity j contains a combinatorial line. It is not hard to see that every subset of <math>[k]^n</math> has complexity at most <math>k-1,</math> so DHJ(k-1,k) is the same as DHJ(k).
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