Ramsey Numbers and Graph Parameters

ording to Ramsey’s Theorem, for any natural p and q there is a minimum number R ( p , q ) such that every graph with at least R ( p , q ) vertices has either a clique of size p or an independent set of size q . In the present paper, we study Ramsey numbers R ( p , q ) for graphs in special classes. It is known that for graphs of bounded co-chromatic number Ramsey numbers are upper-bounded by a linear function of p and q . However, the exact values of R ( p , q ) are known only for classes of graphs of co-chromatic number at most 2. In this paper, we determine the exact values of Ramsey numbers for classes of graphs of co-chromatic number at most 3. It is also known that for classes of graphs of unbounded splitness the value of R ( p , q ) is lower-bounded by (p-1)(q-1)+1 . This lower bound coincides with the upper bound for perfect graphs and for all their subclasses of unbounded splitness. We call a class Ramsey-perfect if there is a constant c such that R(p,q)=(p-1)(q-1)+1 for all p,q≥ c in this class. In the present paper, we identify a number of Ramsey-perfect classes which are not subclasses of perfect graphs.