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The random greedy algorithm for constructing a large partialSteiner-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 eachtriangle 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, orequivalently the time it takes the process to conclude. The intuition that theedge distribution is roughly uniform at all times led to a folklore conjecturethat the final number of edges is $n^{3/2+o(1)}$ with high probability, whereasthe best known upper bound is $n^{7/4+o(1)}$. It is no coincidence that variousmethods break precisely at the exponent 7/4 as it corresponds to the inherentbarrier where co-degrees become comparable to the variations in their valuesthat arose earlier in the process.  In this work we significantly improve upon the previous bounds byestablishing 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 controlkey graph parameters, where the crucial new idea is to harness theself-correcting nature of the process in order to control these parameters wellbeyond the point where their early variation matches the order of theirexpectation.

Random greedy triangle-packing beyond the 7/4 barrier