Threshold Ramsey multiplicity and extremal colorings for odd cycles

The Ramsey number $r(H)$ of a graph $H$ is the minimum positive integer $n$ such that every red/blue edge-coloring of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ of $H$ is the minimum number of monochromatic copies of $H$ over all red/blue edge-colorings of $K_{r(H)}$. The only family for which $m(H)$ has been determined is stars, due to Harary and Prins (1974). Let $C_k$ be a cycle on $k$ vertices. Conlon, Fox, Sudakov, and Wei (2022) conjectured that $m(C_k)=(k-1)!\mathrm{/}2$ for every sufficiently large odd integer $k$. In this paper, we confirm the conjecture and characterize all the extremal colorings of $K_{2k-1}$. This is also the second family for which $m(H)$ has been determined.