Zixiang Xu (徐子翔), Multilinear polynomial methods and stability results on set systems

In 1966, Kleitman established that if $|A\mathrm{△}B|\le d$ for any $A,B\in \mathcal{F}$, then $|\mathcal{F}|\le \sum _{i=0}^k(\genfrac{}{}{0ex}{}{n}{i})$ for $d=2k$, and $|\mathcal{F}|\le 2\sum _{i=0}^k(\genfrac{}{}{0ex}{}{n-1}{i})$ for $d=2k+1$. These upper bounds are attained by the radius-$k$ Hamming ball $\mathcal{K}(n,k):=\{F:F\subseteq [n],|F|\le k\}$ in the even case, and by the family ${\mathcal{K}}_y(n,k):=\{F:F\subseteq [n],|F\setminus \{y\}|\le k\}$ in the odd case. In 2017, Frankl provided a combinatorial proof of a stability result for Kleitman’s theorem, offering improved upper bounds for $|\mathcal{F}|$ when $\mathcal{F}$ is not the extremal structure.

In this talk, I will begin by demonstrating the application of multilinear polynomial methods in extremal set theory, highlighting some interesting techniques. I will then present an algebraic proof of the stability result for Kleitman’s theorem. Finally, I will discuss further applications and explore how to employ linear algebra methods more effectively and flexibly.

This talk is based on joint work with Jun Gao and Hong Liu.