Ramsey Numbers and Graph Parameters

Abstract

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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p-1)(q-1)+1$$\\end{document}. 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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(p,q)=(p-1)(q-1)+1$$\\end{document} for all p,q≥c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p,q\\ge c$$\\end{document} in this class. In the present paper, we identify a number of Ramsey-perfect classes which are not subclasses of perfect graphs.