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DTSTART:20210101T000000
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DTSTART;TZID=Asia/Seoul:20220906T163000
DTEND;TZID=Asia/Seoul:20220906T173000
DTSTAMP:20260420T034129
CREATED:20220719T105944Z
LAST-MODIFIED:20240707T074750Z
UID:5974-1662481800-1662485400@dimag.ibs.re.kr
SUMMARY:Bjarne Schülke\, A local version of Katona's intersection theorem
DESCRIPTION:Katona’s intersection theorem states that every intersecting family $\mathcal F\subseteq[n]^{(k)}$ satisfies $\vert\partial\mathcal F\vert\geq\vert\mathcal F\vert$\, where $\partial\mathcal F=\{F\setminus x:x\in F\in\mathcal F\}$ is the shadow of $\mathcal F$.\nFrankl conjectured that for $n>2k$ and every intersecting family $\mathcal F\subseteq [n]^{(k)}$\, there is some $i\in[n]$ such that $\vert \partial \mathcal F(i)\vert\geq \vert\mathcal F(i)\vert$\, where $\mathcal F(i)=\{F\setminus i:i\in F\in\mathcal F\}$ is the link of $\mathcal F$ at $i$. \nHere\, we prove this conjecture in a very strong form for $n> \binom{k+1}{2}$. \nIn particular\, our result implies that for any $j\in[k]$\, there is a $j$-set $\{a_1\,\dots\,a_j\}\in[n]^{(j)}$ such that \[ \vert \partial \mathcal F(a_1\,\dots\,a_j)\vert\geq \vert\mathcal F(a_1\,\dots\,a_j)\vert.\]A similar statement is also obtained for cross-intersecting families.
URL:https://dimag.ibs.re.kr/event/2022-09-06/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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