Borys Kuca

Recent developments in the polynomial Szemeredi theorem

The polynomial Szemeredi theorem of Bergelson and Leibman guarantees the existence of a large variety of polynomial patterns in large subsets of integers. Originally proof using the soft methods of ergodic theory, it still awaits a quantitative proof that would give bounds for the size of finite sets lacking given progressions. I will report on the progress in the last few years towards this question and its generalisations.