Path Ramsey Numbers and Monochromatic Paths

The path Ramsey number $R(P_m,P_n)$ — the least $N$ so every red/blue coloring of $K_N$ has a red $P_m$ or blue $P_n$ — has the closed form $R(P_m,P_n)=m+\lfloor n/2\rfloor-1$ for $m\ge n\ge 2$; e.g. $R(P_4,P_4)=5$.