The Ramsey number of mixed-parity cycles III

Denote by $R(G_1,G_2,G_3)$ the minimum integer $N$ such that any three-colouring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of a graph $G_i$ coloured with colour $i$ for some $i\in 1,2,3$. In a series of three papers of which this is the third, we consider the case where $G_1,G_2$ and $G_3$ are cycles of mixed parity. Specifically, in this in this paper, we consider $R(C_n,C_m,C_{\mathrm{\ell }})$, where $n$ is even and $m$ and $\mathrm{\ell }$ are odd. Figaj and \L uczak determined an asymptotic result for this case, which we improve upon to give an exact result. We prove that for $n,m$ and $\mathrm{\ell }$ sufficiently large $R(C_n,C_m,C_{\mathrm{\ell }})=\max \{4n-3,n+2m-3,n+2\mathrm{\ell }-3\}$.