Threshold Ramsey multiplicity for paths and even cycles

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