  The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any
two-coloring of the edges of the complete graph $K_n$ contains a monochromatic
copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum
number of monochromatic copies of $H$ taken over all two-edge-colorings of
$K_{r(H)}$. The study of this concept was first proposed by Harary and Prins
almost fifty years ago. In a companion paper, the authors have shown that there
is a positive constant $c$ such that the threshold Ramsey multiplicity for a
path or even cycle with $k$ vertices is at least $(ck)^k$, which is tight up to
the value of $c$. Here, using different methods, we show that the same result
also holds for odd cycles with $k$ vertices.
