  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 first, we
consider the case where $G_1, G_2$ and $G_3$ are cycles of mixed parity.
Specifically, in this and the subsequent 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
$R(C_n,C_m,C_\ell)=\max\{2n+m-3, n+2m-3, \tfrac{1}{2} n +\tfrac{1}{2} m + \ell
- 2\}$. In the case that the longest cycle is of even length, the proof in this
paper is self-contained. 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 the subsequent paper.
