In 1966, Kleitman established that if \( |A \triangle B| \leq d \) for any \( A, B \in \mathcal{F} \), then \( |\mathcal{F}| \leq \sum_{i=0}^{k} \binom{n}{i} \) for \( d = 2k \), and \( |\mathcal{F}| \leq 2 \sum_{i=0}^{k} \binom{n-1}{i} \) for \( d = 2k+1 \). These upper bounds are attained by the radius-\(k\) Hamming ball \( \mathcal{K}(n, k) := \{ F : F \subseteq [n], |F| \leq k \} \) in the even case, and by the family \( \mathcal{K}_y(n, k) := \{ F : F \subseteq [n], |F \setminus \{y\}| \leq k \} \) 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.
