## Combinatorics Seminar

When: Sunday, January 12, 10am
Where: Schreiber 309
Speaker: Eyal Lubetzky, Microsoft Research Redmond
Title: Random Triangle Removal
## Abstract:

Starting from a complete graph on n vertices, repeatedly delete
the edges of a uniformly chosen triangle. This stochastic process
terminates once it arrives at a triangle-free graph, and
the fundamental question is to estimate the final number of edges
(equivalently, the time it takes the process to finish, or how many
edge-disjoint triangles are packed via the random greedy algorithm).
Bollobas and Erdos (1990) conjectured that the expected final number
of edges has order n^{3/2}. An upper bound of o(n^2) was shown by
Spencer (1995) and independently by Rodl and Thoma (1996). Several
bounds were given for variants and generalizations (e.g., Alon, Kim
and Spencer (1997) and Wormald (1999)), while the best known upper
bound for the original question of Bollobas and Erdos was
n^{7/4+o(1)} due to Grable (1997). No nontrivial lower bound was
available.
Here we prove that with high probability the final number of edges
in random triangle removal is equal to n^{3/2+o(1)}, thus
confirming the 3/2 exponent conjectured by Bollobas and Erdos and
matching the predictions of Spencer et al. For the upper bound, for
any fixed \epsilon>0 we construct a family of \exp(O(1/\epsilon))
graphs by gluing O(1/\epsilon) triangles sequentially in
a prescribed manner, and dynamically track all homomorphisms from
them, rooted at any vertex, up to the point where n^{3/2+\epsilon}
edges remain. A system of martingales establishes concentration
for these random variables around their analogous means in
a random graph with corresponding edge density, and a key role is
played by the self-correcting nature of the process. The lower
bound builds on the estimates at that very point to show that
the process will typically terminate with at least n^{3/2-o(1)}
edges left.
Joint work with Tom Bohman and Alan Frieze.