  We prove that in every $2$-edge-colouring of $K_n$ there is a collection of
$n^2/12 + o(n^2)$ edge-disjoint monochromatic triangles, thus confirming a
conjecture of Erd\H{o}s. We also prove a corresponding stability result,
showing that $2$-colourings that are close to attaining the aforementioned
bound have a colour class which is close to bipartite. As part of our proof, we
confirm a recent conjecture of Tyomkyn about the fractional version of this
problem.
