Progression-Free Sets in Arithmetic Sequences

Explore a 48-minute conference talk by Péter Pál Pach at BIMSA on progression-free sets in the context of the International Conference on Bounded Subsets (ICBS) 2024. Delve into the discussion of lower and upper bounds for the maximal possible size of sets avoiding nontrivial k-term arithmetic progressions in Z_m^n, specifically focusing on r_k(Z_m^n) in certain cases. Discover how this quantity is proven to be exponentially smaller than m^n when k=3≤m, achieved through a novel variant of the polynomial method. Learn about the successful applications of this technique to various problems in the field of combinatorics and number theory.