The classical Ramsey theory
for graphs studies the Ramsey numbers r(m,n). This is the smallest p such
that every 2-coloring of the lines of the complete graph Kp contains a green
Km or a red Kn. 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 Kp contains a
monochromatic F. Here we introduce the off-diagonal numbers: 7⋅(F1,F2) with
F1≠F2 is the minimum p such that every 2-coloring of Kp contains a green F1
or a red F2. With the help of a general lower bound, the exact values of
r(F1,F2) are determined for all graphs Fi with less than five points having no
isolates.
