Minimum-degree Ramsey threshold for monochromatic triangle tilings

Determine whether every 2-edge-coloured graph G on n≥25 vertices contains a monochromatic copy of mK3 with m given by the three-case formula in Question 1.6: m=⌊(δ(G)+1)/5⌋ when 15n/17≤δ(G), m=⌊(4δ(G)−3n+1)/3⌋ when 6n/7≤δ(G)≤15n/17, and m=⌈(5δ(G)−4n)/2⌉ when 4n/5≤δ(G)≤6n/7.

Background

The paper studies Ramsey-type tiling problems in 2-edge-coloured dense graphs, where the host graph is not necessarily complete but has prescribed minimum degree. Theorem 1.5 determines the largest guaranteed monochromatic triangle tiling in two ranges of the minimum degree, but leaves a gap for minimum degrees between 5n/6 and 15n/17.

Question 1.6 proposes a complete three-range formula for the guaranteed number of vertex-disjoint monochromatic triangles. Extremal constructions in Section 2.2 show that the proposed bounds cannot be increased. An affirmative answer would extend the Burr–Erdős–Spencer theorem for monochromatic triangle tilings from complete graphs to the minimum-degree setting.