This talk deals with generalizations of the classic finite Ramsey theorem obtained by substituting "set of cardinality n" with some notion of "large set". The prototype of these results is of course the statement that Paris and Harrington showed unprovable in PA in 1977. Since then several extensions were considered, typically using notions of α-largeness for α an ordinal below ε_0. Our results extend this approach by dealing with notions of largeness captured by barriers (one of the combinatorial ingredients of better quasi-order theory). We use these largeness notions almost everywhere in the statements (the only cardinality left is the number of colors). Quite surprisingly, in many cases we obtain tight bounds on these "generalized Ramsey numbers", in contrast with the classic finite case where tight bounds are known only for very few cases involving very small numbers.
This is joint work with Antonio Montalban and Andrea Volpi