기초과학연구원 이산수학그룹
On April 11, 2022, Younjin Kim (김연진) from POSTECH gave a talk at the Discrete Math Seminar on the number of k-dimensional corner-free subsets of the k-dimensional grid {1,2,…n}^k. The title of her talk was “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 $|A| \geq \delta N$ 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.