On the number of triangles in K₄-free graphs

Erdős asked whether for any n-vertex graph G, the parameter p^* (G) = ₈ ₁ (|V (Gᵢ) |-1) is at most n²/4, where the minimum is taken over all edge decompositions of G into edge-disjoint cliques Gᵢ. In a restricted case (also conjectured independently by Erdős), Győri and Keszegh Combinatorica, 37 (6) (2017), 1113--1124 proved that p^* (G) n²/4 for all K₄-free graphs G. Motivated by their proof approach, they conjectured that for any n-vertex K₄-free graph G with e edges, and any greedy partition P of G of size r, the number of triangles in G is at least r (e-r (n-r) ). If true, this would imply a stronger bound on p^* (G). In this paper, we disprove their conjecture by constructing infinitely many counterexamples with arbitrarily large gap. We further establish a corrected tight lower bound on the number of triangles in such graphs, which would recover the conjectured bound once some small counterexamples we identify are excluded.