"Large subset" is not a technical term. The right question here is not "how is large subset" defined; it's "What does Ramsey's theorem actually state?" A tota

"Large subset" is not a technical term. The right question here is not "how is large subset" defined; it's "What does Ramsey's theorem actually state?" A totally abstract version of the theorem would be (note: I'm only considering finite versions of Ramsey's theorem here), Ramsey's theorem states that given any n and m, there's a number R(n,m) such that if you have a complete graph on R(n,m) vertices, and you color each edge red or blue, there's either a uniformly red subset of n vertices or a uniformly blue subset of m vertices. A more useful version would give an example of a number you could pick for R(n,m). (Normally we define R(n,m) to be the smallest such number, so you're giving an upper bound on it.) For instance, Erdos and Szekeres proved that R(n,m)<=(r+s-2 choose r-1).