PROVED
This has been solved in the affirmative.
Is it true that in any $2$-colouring of the edges of $K_n$ there must exist at least\[(1+o(1))\frac{n^2}{12}\]many edge-disjoint monochromatic triangles?
Conjectured by Erdős, Faudreee, and Ordman. This would be best possible, as witnessed by dividing the vertices of $K_n$ into two equal parts and colouring all edges between the parts red and all edges inside the parts blue.
The answer is yes, proved by Gruslys and Letzter
[GrLe20].
In
[Er97d] Erdős also asks for a lower bound for the count of edge-disjoint monochromatic triangles in single colour (the colour chosen to maximise this quantity), and speculates that the answer is $\geq cn^2$ for some constant $c>1/24$.
View the LaTeX source
Additional thanks to: Julius Schmerling and Tuan Tran
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #76, https://www.erdosproblems.com/76, accessed 2026-01-14
