Younjin Kim (김연진), On the extremal problems related to Szemerédi’s theorem

In 1975, Szemerédi proved that for every real number $\delta >0$ and every positive integer $k$, there exists a positive integer $N$ such that every subset $A$ of the set $\{1,2,\cdots ,N\}$ with $|A|\ge \delta N$ contains an arithmetic progression of length $k$. There has been a plethora of research related to Szemerédi’s theorem in many areas of mathematics. In 1990, Cameron and Erdős proposed a conjecture about counting the number of subsets of the set $\{1,2,\dots ,N\}$ which do not contain an arithmetic progression of length $k$. In the talk, we study a natural higher dimensional version of this conjecture, and also introduce recent extremal problems related to Szemerédi’s theorem.