Complexity of a set: Difference between revisions

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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>
-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''.
-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).

Complexity of a set: Difference between revisions

Revision as of 20:12, 27 March 2009

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