Graph classes and Ramsey numbers

For a graph class GG and any two positive integers ii and jj, the Ramsey number RG(i,j)RG(i,j) is the smallest positive integer such that every graph in GG on at least RG(i,j)RG(i,j) vertices has a clique of size ii or an independent set of size jj. For the class of all graphs, Ramsey numbers are notoriously hard to determine, and they are known only for very small values of ii and jj. Even if we restrict GG to be the class of claw-free graphs, it is highly unlikely that a formula for determining RG(i,j)RG(i,j) for all values of ii and jj will ever be found, as there are infinitely many nontrivial Ramsey numbers for claw-free graphs that are as difficult to determine as for arbitrary graphs. Motivated by this difficulty, we establish here exact formulas for all Ramsey numbers for three important subclasses of claw-free graphs: line graphs, long circular interval graphs, and fuzzy circular interval graphs. On the way to obtaining these results, we also establish all Ramsey numbers for the class of perfect graphs. Such positive results for graph classes are rare: a formula for determining RG(i,j)RG(i,j) for all values of ii and jj, when GG is the class of planar graphs, was obtained by Steinberg and Tovey (1993), and this seems to be the only previously known result of this kind. We complement our aforementioned results by giving exact formulas for determining all Ramsey numbers for several graph classes related to perfect graphs.