Long monochromatic even cycles in 3-edge-coloured graphs of large minimum degree

We show that for every $\eta >0$, there exists $n_0$ such that for every even $n$, $n\ge n_0$, and every graph $G$ with $(2+\eta )n$ vertices and minimum degree at least $(7\mathrm{/}4+4\eta )n$, each colouring of the edges of $G$ with three colours results in a monochromatic cycle of length $n$.