Random greedy triangle-packing beyond the 7/4 barrier

The random greedy algorithm for constructing a large partial Steiner-Triple-System is defined as follows. Begin with a complete graph on $n$ vertices and proceed to remove the edges of triangles one at a time, where each triangle removed is chosen uniformly at random out of all remaining triangles. This stochastic process terminates once it arrives at a triangle-free graph, and a longstanding open problem is to estimate the final number of edges, or equivalently the time it takes the process to conclude. The intuition that the edge distribution is roughly uniform at all times led to a folklore conjecture that the final number of edges is $n^{3/2+o(1)}$ with high probability, whereas the best known upper bound is $n^{7/4+o(1)}$. It is no coincidence that various methods break precisely at the exponent 7/4 as it corresponds to the inherent barrier where co-degrees become comparable to the variations in their values that arose earlier in the process. In this work we significantly improve upon the previous bounds by establishing that w.h.p. the number of edges in the final graph is at most $ n^{5/3+o(1)} $. Our approach relies on a system of martingales used to control key graph parameters, where the crucial new idea is to harness the self-correcting nature of the process in order to control these parameters well beyond the point where their early variation matches the order of their expectation.

💡 Research Summary

The paper studies the random greedy algorithm for triangle‑packing: starting from the complete graph on (n) vertices, at each step a triangle is chosen uniformly at random among all remaining triangles and its three edges are deleted. The process stops when the graph becomes triangle‑free; the number of remaining edges (|E(M)|) (or equivalently the number of removed triangles (M)) is the main object of interest.

Historically, the folklore conjecture (attributed to Folk­lore) predicts (|E(M)| = n^{3/2+o(1)}) with high probability, based on the heuristic that the evolving graph behaves like an Erdős–Rényi random graph of the same edge density. However, rigorous analysis has been stuck at the exponent (7/4). The barrier appears when the co‑degree (|N_{u,v}|) (the number of common neighbours of a pair ((u,v))) has a standard deviation comparable to its mean; this happens roughly when the remaining edge density (p) satisfies (p\approx n^{-1/4}), i.e., when about (n^{7/4}) edges are left. At that point, earlier techniques lose control over co‑degrees, and the analysis collapses.

The authors break this barrier by introducing a sophisticated martingale framework that exploits the self‑correcting nature of the process. They track an ensemble of random variables simultaneously:

• (Q(i)): the number of triangles in the current graph (G(i));
• (Y_{u,v}(i)=|N_{u,v}(i)|): co‑degree of a vertex pair;
• (R_{u,v}(i)): a refined count of ordered pairs ((x,y)) with (xy) an edge, (x) a common neighbour of (u) and (v), and (y) a neighbour of (u) (excluding (v));
• (Y_u(i)): degree of vertex (u);
• (T_u(i)): the number of edges inside the neighbourhood of (u);
• (Y_{u,v,w}(i)): triple‑intersection size.

For each variable they compute the exact one‑step expected change (e.g., (\mathbb{E}