Georgia Institute of Technology College of Sciences

The Ramsey number R(3,k) is the smallest n so that every triangle free graph on n vertices has an independent set of size k.  Upper bounds to R(3,k) consist of finding independent sets in triangle free graphs, while lower bounds consist of constructing triangle free graphs with no large independent set.  The previous best-known lower bound was independently due to works of Bohman-Keevash and Fiz Pontiveros-Griffiths-Morris, both of which analyzed the triangle-free process.  The analysis of the triangle-free process is a delicate dance of demonstrating self-correction and requires tracking a large family of graph statistics simultaneously.  We will discuss a new lower bound to R(3,k) and provide a gentle introduction to the concept of "the Rodl nibble," with an emphasis on which ideas simplify our analysis.  This is based on joint work with Marcelo Campos, Matthew Jenssen and Julian Sahasrabudhe.