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 {\displaystyle R(s,t):s,t\in \mathbb {N} } is defined as the least integer n such that whenever the complete graph K n {\displaystyle K_{n}} is coloured, such that each edge is one of two colours, it contains either a  K s {\displaystyle K_{s}} with all edges the first colour, or a K t {\displaystyle K_{t}} 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 ) {\displaystyle R(s,t)} exists  ∀ s , t ∈ N {\displaystyle \forall s,t\in \mathbb {N} }