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X-ORIGINAL-URL:https://dimag.ibs.re.kr
X-WR-CALDESC:Events for Discrete Mathematics Group
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TZID:Asia/Seoul
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TZOFFSETFROM:+0900
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DTSTART:20230101T000000
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DTSTART;TZID=Asia/Seoul:20241223T163000
DTEND;TZID=Asia/Seoul:20241223T173000
DTSTAMP:20260417T234205
CREATED:20241115T065846Z
LAST-MODIFIED:20241214T083034Z
UID:10184-1734971400-1734975000@dimag.ibs.re.kr
SUMMARY:Zixiang Xu (徐子翔)\, Multilinear polynomial methods and stability results on set systems
DESCRIPTION: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. \nIn 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. \nThis talk is based on joint work with Jun Gao and Hong Liu.
URL:https://dimag.ibs.re.kr/event/2024-12-23/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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