Georgia Institute of Technology College of Sciences

Fix a graph or uniform hypergraph $F$ and consider the following simple 
process. Start with the complete (hyper)graph $K_n$ on $n$ vertices and 
repeatedly remove the edge set of a copy of $F$, chosen uniformly at 
random among all copies present at that time. Stop when no copy of $F$ 
is left. This is the $F$-removal process, and the key question is when 
it ends, that is, how many edges are left at the end.

This question has proven to be very hard. In 2015, Bohman, Frieze and 
Lubetzky famously proved that the triangle-removal process typically 
ends with $n^{3/2+o(1)}$ edges. Until now, the answer was not known for 
any other $F$.

We determine when the $F$-removal process ends for many choices of $F$. 
This includes cliques, where it confirms a well-known conjecture of 
Bennett and Bohman on the final number of edges. This is joint work with 
Kühn.