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

## Zimin words

## 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.