Ramsey Numbers and Graph Parameters

According 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 \(p,q\ge c\) in this class. In the present paper, we identify a number of Ramsey-perfect classes which are not subclasses of perfect graphs.