Ramsey theory

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