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  $$ \forall s,t\in \mathbb N$$