I heard a lecture by Benny Sudakov on the remarkable paper Tower-type bounds for unavoidable patterns in words, by David Conlon, Jacob Fox, and Benny Sudakov. Here are the slides, and let me let the slides speak for themselves.
The problem and what was known: upper bounds
Earlier lower bounds
The new results
The proof is not difficult. It uses a probabilistic argument based on the Lovasz local lemma, and a step up argument.