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 $R(C_n,C_m,C_{\ell})$, where $n$ is even and $m$ and $\ell$ 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 $\ell$ sufficiently large $R(C_n,C_m,C_\ell)=\max\{4n-3, n+2m-3, n+2\ell-3\}$.