  The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that
any two-coloring of the edges of the complete graph $K_n$ contains a
monochromatic copy of $H$. While this definition only asks for a single
monochromatic copy of $H$, it is often the case that every two-edge-coloring of
the complete graph on $r(H)$ vertices contains many monochromatic copies of
$H$. The minimum number of such copies over all two-colorings of $K_{r(H)}$
will be referred to as the threshold Ramsey multiplicity of $H$. Addressing a
problem of Harary and Prins, who were the first to systematically study this
quantity, we show that there is a positive constant $c$ such that the threshold
Ramsey multiplicity of a path or an even cycle on $k$ vertices is at least
$(ck)^k$. This bound is tight up to the constant $c$. We prove a similar result
for odd cycles in a companion paper.
