Ramsey Number Bounds

Ramsey's theorem implies that for any given natural numbers $k$ and $l$, there exists some $R(k,l)$ such that a graph having at least $R(k,l)$ vertices must have either a clique of cardinality $k$ or an anticlique (independent set) of cardinality $l$. Equivalently, for a complete graph of size $R(k,l)$, every red/blue colouring of the edges must yield an entirely red $k$-clique or an entirely blue $l$-clique. Although $R(k,l)$ is for practical purposes impossible to calculate from $k$ and $l$, some upper and lower bounds are known. The celebrated probabilistic argument by Paul Erdős is formalised here, with various of its consequences.