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Ramsey's theorem (Theorem)

Ramsey's theorem states that a particular arrows relation,

$\displaystyle \omega\rightarrow(\omega)^n_k$

holds for any integers $ n$ and $ k$.

In words, if $ f$ is a function on sets of integers of size $ n$ whose range is finite then there is some infinite $ X\subseteq\omega$ such that $ f$ is constant on the subsets of $ X$ of size $ n$.

As an example, consider the case where $ n=k=2$, and $ f\colon [\omega]^2\rightarrow\{0,1\}$ is defined by:

\begin{displaymath}f(\{x,y\})=\left\{ \begin{array}{ll} 1&\text{ if ~} x=y^2 \text{ or ~} y=x^2\ 0&\text{otherwise} \end{array}\right.\end{displaymath}

Then let $ X\subseteq\omega$ be the set of integers which are not perfect squares. This is clearly infinite, and obviously if $ x,y\in X$ then neither $ x=y^2$ nor $ y=x^2$, so $ f$ is homogeneous on $ X$.



"Ramsey's theorem" is owned by mathcam. [ owner history (1) ]
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See Also: Ramsey's theorem, graph theory


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proof of Ramsey's theorem (Proof) by mathcam
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Cross-references: homogeneous, perfect squares, subsets, infinite, finite, range, size, function, integers, arrows relation
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This is version 6 of Ramsey's theorem, born on 2002-08-10, modified 2007-06-17.
Object id is 3285, canonical name is RamseysTheorem.
Accessed 3958 times total.

Classification:
AMS MSC: 05D10 (Combinatorics :: Extremal combinatorics :: Ramsey theory)
Pending Errata and Addenda
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Rewrite by Henry on 2007-06-17 14:51:19
This entry could stand a complete rewrite, probably including a merger with the other entry on Ramsey's theorem. (I kept putting off that correction hoping to getting aroudn to doing it.)

It would be nice to give both the finitary and infinitary versions, some more exposition about what the theorem means (for instance the traditional "friends versus strangers at a party" example), and the discussion about Ramsey's numbers from the other version.
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