Many disjoint triangles in co-triangle-free graphs

We prove that any $n$-vertex graph whose complement is triangle-free contains $n^2\mathrm{/}12-o(n^2)$ edge-disjoint triangles. This is tight for the disjoint union of two cliques of order $n\mathrm{/}2$. We also prove a corresponding stability theorem, that all large graphs attaining the above bound are close to being bipartite. Our results answer a question of Alon and Linial, and make progress on a conjecture of Erd\H{o}s.