  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.
