The Ramsey number of mixed-parity cycles II

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 second, we consider the case where $G_1, G_2$ and $G_3$ are cycles of mixed parity. Here and in the previous paper, we consider $R(C_n,C_m,C_{\ell})$, where $n$ and $m$ are even and $\ell$ is 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 $\ell$ sufficiently large $R(C_n,C_m,C_\ell)=\max\{2n+m-3, n+2m-3, \tfrac{1}{2} n +\tfrac{1}{2} m + \ell - 2\}$. The proof of this result is mostly contained within the first paper in this series, however, in the case that the longest cycle is of odd length, we require an additional technical result, the proof of which makes up the majority of this paper.