  This paper defines, for each graph $G$, a flag vector $fG$. The flag vectors
of the graphs on $n$ vertices span a space whose dimension is $p(n)$, the
number of partitions on $n$. The analogy with convex polytopes indicates that
the linear inequalities satisfied by $fG$ may be both interesting and
accessible. Such would provide inequalities both sharp and subtle on the
combinatorial structure of $G$. These may be related to Ramsey theory.
