arrows relation
Let , that is, the set of subsets of of size . Then given some cardinals , , and
states that for any set of size and any function , there is some and some such that and for any , .
In words, if is a partition of into subsets then is constant on a subset of size (a homogeneous subset).
As an example, the pigeonhole principle is the statement that if is finite and then:
That is, if you try to partition into fewer than pieces then one piece has more than one element.
Observe that if
then the same statement holds if:
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is made larger (since the restriction of to a set of size can be considered)
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is made smaller (since a subset of the homogeneous set will suffice)
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is made smaller (since any partition into fewer than pieces can be expanded by adding empty sets to the partition)
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is made smaller (since a partition of where can be extended to a partition of by where is the smallest elements of )
is used to state that the corresponding relation is false.
References
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Jech, T. Set Theory, Springer-Verlag, 2003
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Just, W. and Weese, M. Topics in Discovering Modern Set Theory, II, American Mathematical Society, 1996
Title | arrows relation |
Canonical name | ArrowsRelation |
Date of creation | 2013-03-22 17:48:54 |
Last modified on | 2013-03-22 17:48:54 |
Owner | Henry (455) |
Last modified by | Henry (455) |
Numerical id | 5 |
Author | Henry (455) |
Entry type | Definition |
Classification | msc 05A18 |
Classification | msc 03E05 |
Related topic | PartitionsLessThanCofinality |
Related topic | ErdosRadoTheorem |
Defines | homogeneous |
Defines | arrows |
Defines | homogeneous set |
Defines | homogeneous subset |