Carlson-Simpson theorem: Difference between revisions
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'''Carlson-Simpson theorem''' (k=3): If <math>[3]^\omega := \bigcup_{n=0}^\infty [3]^n</math> is partitioned into finitely many color classes, then one of the color classes contains an infinite-dimensional [[combinatorial subspace]], i.e. another copy of <math>[3]^\omega</math>. | '''Carlson-Simpson theorem''' (k=3): If <math>[3]^\omega := \bigcup_{n=0}^\infty [3]^n</math> is partitioned into finitely many color classes, then one of the color classes contains an infinite-dimensional [[combinatorial subspace]], i.e. another copy of <math>[3]^\omega</math>. | ||
Implies the [[coloring Hales-Jewett theorem]]. | Implies the [[coloring Hales-Jewett theorem]]. | ||
The Carlson-Simpson theorem and the [[Graham-Rothschild theorem]] have a common generalisation, [[Carlson's theorem]]. | The Carlson-Simpson theorem and the [[Graham-Rothschild theorem]] have a common generalisation, [[Carlson's theorem]]. | ||
Both the Carlson-Simpson theorem and Carlson's theorem are is used in the [[Furstenberg-Katznelson argument]]. | Both the Carlson-Simpson theorem and Carlson's theorem are is used in the [[Furstenberg-Katznelson argument]]. | ||
Latest revision as of 23:52, 15 February 2009
Carlson-Simpson theorem (k=3): If [math]\displaystyle{ [3]^\omega := \bigcup_{n=0}^\infty [3]^n }[/math] is partitioned into finitely many color classes, then one of the color classes contains an infinite-dimensional combinatorial subspace, i.e. another copy of [math]\displaystyle{ [3]^\omega }[/math].
Implies the coloring Hales-Jewett theorem.
The Carlson-Simpson theorem and the Graham-Rothschild theorem have a common generalisation, Carlson's theorem.
Both the Carlson-Simpson theorem and Carlson's theorem are is used in the Furstenberg-Katznelson argument.
