Elimination of the asymptotic error term for dense mixed-colour triangle tilings

Prove that the o(n) error term in case (M.2) of Theorem 1.7 can be removed, thereby establishing an exact lower bound of ⌊(4δ(G)−3n)/2⌋ monochromatic copies of K3 in a K3-tiling of every sufficiently large 2-edge-coloured n-vertex graph G with 5n/6≤δ(G)≤7n/8.

Background

Theorem 1.7 gives an asymptotic result for K3-tilings in which each triangle is monochromatic but different triangles may have different colours. In the range 5n/6≤δ(G)≤7n/8, the theorem guarantees at least ⌊(4δ(G)−3n)/2⌋−o(n) triangles, and the authors show this is best possible up to the o(n) term.

The paper explicitly conjectures that the error term is unnecessary. Removing it would sharpen the result from an asymptotic guarantee to an exact one and would likely require methods avoiding the loss introduced by the regularity and blow-up arguments used in the proof.