  We prove that every $n$-vertex graph with at least $\binom{n}{2} - (n - 4)$
edges has a fractional triangle decomposition, for $n \ge 7$. This is a key
ingredient in our proof, given in a companion paper, that every $n$-vertex
$2$-coloured complete graph contains $n^2/12 + o(n^2)$ edge-disjoint
monochromatic triangles, which confirms a conjecture of Erd\H{o}s.
