Ramsey theory

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This editable Main Article is under development and subject to a disclaimer.

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Ramsey Theory is a branch of Graph theory which studies seemingly orderless systems and analyses the conditions under which order must be present. Typically it makes statements about the existence of specific sub-graphs in arbitrary graphs.

## Ramsey Numbers[edit]

The Ramsey Number R(s,t):s,t∈ℕ is defined as the least integer n such that whenever the complete graph Kn is coloured, such that each edge is one of two colours, it contains either a  Ks with all edges the first colour, or a Kt with all edges the second colour, if such an n exists.

Ramsey Numbers are rarely calculated, largely due to the rate at which the complexity of verifying the results grows.

### Ramsey's Theorem[edit]

Ramsey's Theorem states that R(s,t) exists  ∀s,t∈ℕ