Generalized Ramsey theory for graphs. III. Small off-diagonal numbers
The classical Ramsey theory for graphs studies the Ramsey numbers r(m, n). This is the smallest p such that every 2coloring of the lines of the complete graph K p contains a green K m or a red K n In the preceding papers in this series, we developed the theory and calculation of the diagonal numbers r{F) for a graph F with no isolated points, as the smallest p for which every 2-coloring of K p contains a monochromatic F Here we introduce the off-diagonal numbers: r(Fu F 2 ) with JF F 2 is the minimum p such that every 2coloring of K v contains a green F\ or a red F 2 . With the help of a general lower bound, the exact values of r(F lf F 2 ) are determined for all graphs Fi with less than five points having no isolates.