# Difference between revisions of "Hindman's theorem"

From Polymath1Wiki

Line 1: | Line 1: | ||

'''Hindman's theorem''': If <math>[2]^\omega := \bigcup_{n=0}^\infty [2]^n</math> is finitely colored, then one of the color classes contain all elements of an infinite-dimensional [[combinatorial subspace]] which contain the digit 1, and such that none of the fixed digits of this subspace are equal to 1. | '''Hindman's theorem''': If <math>[2]^\omega := \bigcup_{n=0}^\infty [2]^n</math> is finitely colored, then one of the color classes contain all elements of an infinite-dimensional [[combinatorial subspace]] which contain the digit 1, and such that none of the fixed digits of this subspace are equal to 1. | ||

− | The generalization of this theorem | + | The generalization of this theorem which replaces 2 with larger k is [[Carlson's theorem]]. Hindman's theorem also implies [[Folkman's theorem]]. |

## Latest revision as of 04:06, 21 May 2009

**Hindman's theorem**: If [math][2]^\omega := \bigcup_{n=0}^\infty [2]^n[/math] is finitely colored, then one of the color classes contain all elements of an infinite-dimensional combinatorial subspace which contain the digit 1, and such that none of the fixed digits of this subspace are equal to 1.

The generalization of this theorem which replaces 2 with larger k is Carlson's theorem. Hindman's theorem also implies Folkman's theorem.