Carlson's theorem - Revision history

Teorth at 07:51, 16 February 2009

2009-02-16T07:51:44Z

← Older revision		Revision as of 00:51, 16 February 2009
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	This follows by viewing a combinatorial line in <math>[3]^\omega</math> as an element in <math>[4]^\omega</math> containing at least one 4, thinking of the 4 as the "wildcard" for the line.		This follows by viewing a combinatorial line in <math>[3]^\omega</math> as an element in <math>[4]^\omega</math> containing at least one 4, thinking of the 4 as the "wildcard" for the line.

			The k=2 version of this theorem is [[Hindman's theorem]].

Teorth at 02:14, 16 February 2009

2009-02-16T02:14:19Z

← Older revision		Revision as of 19:14, 15 February 2009
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	'''Carlson~~-Simpson Graham-Rothschild~~ theorem''' (k=3), Version I: If <math>[4]^\omega := \bigcup_{n=0}^\infty [4]^n</math> is partitioned into finitely many color classes, then there exists an infinite-dimensional [[combinatorial subspace]] with no fixed coordinate equal to 4, such that every element of this combinatorial subspace with at least one 4 has the same color.		'''Carlson's theorem''' (k=3), Version I: If <math>[4]^\omega := \bigcup_{n=0}^\infty [4]^n</math> is partitioned into finitely many color classes, then there exists an infinite-dimensional [[combinatorial subspace]] with no fixed coordinate equal to 4, such that every element of this combinatorial subspace with at least one 4 has the same color.

	This theorem is a common generalization of the [[Carlson-Simpson theorem]] and the [[Graham-Rothschild theorem]]. It plays a key role in the [[Furstenberg-Katznelson argument]]. It is necessary to restrict to elements containing at least one 4; consider the coloring that colors a string black if it contains at least one 4, and white otherwise.		This theorem is a common generalization of the [[Carlson-Simpson theorem]] and the [[Graham-Rothschild theorem]]. It plays a key role in the [[Furstenberg-Katznelson argument]]. It is necessary to restrict to elements containing at least one 4; consider the coloring that colors a string black if it contains at least one 4, and white otherwise.
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	It has an equivalent formulation:		It has an equivalent formulation:

	'''Carlson~~-Simpson Graham-Rothschild~~ theorem''' (k=3), Version I: If the [[combinatorial line]]s in <math>[3]^\omega := \bigcup_{n=0}^\infty [3]^n</math> are partitioned into finitely many color classes, then there exists an infinite-dimensional [[combinatorial subspace]] such that all combinatorial lines in this subspace have the same color.		'''Carlson's theorem''' (k=3), Version I: If the [[combinatorial line]]s in <math>[3]^\omega := \bigcup_{n=0}^\infty [3]^n</math> are partitioned into finitely many color classes, then there exists an infinite-dimensional [[combinatorial subspace]] such that all combinatorial lines in this subspace have the same color.

	This follows by viewing a combinatorial line in <math>[3]^\omega</math> as an element in <math>[4]^\omega</math> containing at least one 4, thinking of the 4 as the "wildcard" for the line.		This follows by viewing a combinatorial line in <math>[3]^\omega</math> as an element in <math>[4]^\omega</math> containing at least one 4, thinking of the 4 as the "wildcard" for the line.

Teorth: Carlson-Simpson Graham-Rothschild theorem moved to Carlson's theorem

2009-02-16T02:12:57Z

Carlson-Simpson Graham-Rothschild theorem moved to Carlson's theorem

← Older revision	Revision as of 19:12, 15 February 2009
(No difference)

Teorth: New page: '''Carlson-Simpson Graham-Rothschild theorem''' (k=3), Version I: If $[4]^\omega := \bigcup_{n=0}^\infty [4]^n$ is partitioned into finitely many color classes, then there exist...

2009-02-16T01:36:02Z

New page: '''Carlson-Simpson Graham-Rothschild theorem''' (k=3), Version I: If <math>[4]^\omega := \bigcup_{n=0}^\infty [4]^n</math> is partitioned into finitely many color classes, then there exist...

New page

'''Carlson-Simpson Graham-Rothschild theorem''' (k=3), Version I: If <math>[4]^\omega := \bigcup_{n=0}^\infty [4]^n</math> is partitioned into finitely many color classes, then there exists an infinite-dimensional [[combinatorial subspace]] with no fixed coordinate equal to 4, such that every element of this combinatorial subspace with at least one 4 has the same color.

This theorem is a common generalization of the [[Carlson-Simpson theorem]] and the [[Graham-Rothschild theorem]]. It plays a key role in the [[Furstenberg-Katznelson argument]]. It is necessary to restrict to elements containing at least one 4; consider the coloring that colors a string black if it contains at least one 4, and white otherwise.

It has an equivalent formulation:

'''Carlson-Simpson Graham-Rothschild theorem''' (k=3), Version I: If the [[combinatorial line]]s in <math>[3]^\omega := \bigcup_{n=0}^\infty [3]^n</math> are partitioned into finitely many color classes, then there exists an infinite-dimensional [[combinatorial subspace]] such that all combinatorial lines in this subspace have the same color.

This follows by viewing a combinatorial line in <math>[3]^\omega</math> as an element in <math>[4]^\omega</math> containing at least one 4, thinking of the 4 as the "wildcard" for the line.

Carlson's theorem - Revision history

Teorth at 07:51, 16 February 2009

Teorth at 02:14, 16 February 2009

Teorth: Carlson-Simpson Graham-Rothschild theorem moved to Carlson's theorem

Teorth: New page: '''Carlson-Simpson Graham-Rothschild theorem''' (k=3), Version I: If [4]^\omega := \bigcup_{n=0}^\infty [4]^n is partitioned into finitely many color classes, then there exist...

Teorth: New page: '''Carlson-Simpson Graham-Rothschild theorem''' (k=3), Version I: If $[4]^\omega := \bigcup_{n=0}^\infty [4]^n$ is partitioned into finitely many color classes, then there exist...