Edge count in the triangle-removal process

Prove or disprove the Bollobás–Erdős conjecture that the triangle-removal process on the complete graph with n vertices leaves a triangle-free graph with Θ(n^{3/2}) edges.

Background

The triangle-removal process repeatedly removes a uniformly random triangle from the complete graph until the remaining graph is triangle-free. The paper reports that the best known result establishes only n{3/2+o(1)} remaining edges, which does not settle the conjectured Θ(n{3/2}) order with matching constant-factor bounds.

References

This problem was introduced by Bollob{a}s and Erd\H{o}s who conjectured that the remaining graph has $\Theta(n{3/2})$ edges. The best known result towards this conjecture is due to Bohman, Frieze and Lubetzky , who proved the triangle-free graph that results from the triangle-removal process has $n{3/2+o(1)}$ edges.

— A new lower bound for the Ramsey numbers $R(3,k)$  (2505.13371 - Campos et al., 19 May 2025) in Section 1, subsection “A history of the problem”