Recent developments in the polynomial Szemerédi theorem

The polynomial Szemerédi theorem of Bergelson and Leibman asserts the existence of a broad class of polynomial patterns in large subsets of integers. Originally proved using soft methods of ergodic theory, it still lacks a quantitative proof that would give bounds for the size of finite sets avoiding  a given poltnomial progression. In recent years, there has been a lot of progress on obtaining such bounds for varies classes of configurations, which I will report on in this talk.