Ramsey theory

Ramsey Theory is 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

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

Ramsey's Theorem states that   $R(s,t)$  exists  $\forall s,t\in \mathbb {N}$