The textbook that I'm using to learn combinatorial set theory, Combinatorial Set Theory With A Gentle Introduction To Forcing, has this purely set-theoretic definition of The Finite Ramsey Theorem which doesn't appear anywhere else I search (all that pops up is the Graph Theory definition). I just want to understand how this definition actually pertains to the existence of a Ramsey Number for every set of two natural numbers. The definition they provide is as follows:
COROLLARY 2.3 (FINITE RAMSEY THEOREM). For all m,n,r ∈ ω, where r ≥ 1 and n ≤ m, there exists an N ∈ ω, where N ≥ m, such that for every colouring of [N]^n with r colours, there exists a set H ∈ N^m, all of whose n-element subsets have the same colour.