A simple argument of Erdos shows that every set of integers has a subset of relative density at least 1/3 that is sum-free, i.e. contains no solutions to x+y=z. He conjectured that the constant 1/3 is best possible.
This conjecture was recently proved by Sean Eberhard, Ben Green and the speaker. A key component of the proof is a structural result concerning sets of integers with doubling constant strictly less than 4.
We will attempt to outline the proof of the sum-free statement, with an emphasis on the role of this doubling 4 lemma.