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DTSTART:20210101T000000
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DTSTART;TZID=Asia/Seoul:20220321T163000
DTEND;TZID=Asia/Seoul:20220321T173000
DTSTAMP:20260423T172723
CREATED:20220321T073000Z
LAST-MODIFIED:20240707T080150Z
UID:5277-1647880200-1647883800@dimag.ibs.re.kr
SUMMARY:Jaehoon Kim (김재훈)\, Ramsey numbers of cycles versus general graphs
DESCRIPTION:The Ramsey number $R(F\,H)$ is the minimum number $N$ such that any $N$-vertex graph either contains a copy of $F$ or its complement contains $H$. Burr in 1981 proved a pleasingly general result that for any graph $H$\, provided $n$ is sufficiently large\, a natural lower bound construction gives the correct Ramsey number involving cycles: $R(C_n\,H)=(n-1)(\chi(H)-1)+\sigma(H)$\, where $\sigma(H)$ is the minimum possible size of a colour class in a $\chi(H)$-colouring of $H$. Allen\, Brightwell and Skokan conjectured that the same should be true already when $n\geq |H|\chi(H)$. \nWe improve this 40-year-old result of Burr by giving quantitative bounds of the form $n\geq C|H|\log^4\chi(H)$\, which is optimal up to the logarithmic factor. In particular\, this proves a strengthening of the Allen-Brightwell-Skokan conjecture for all graphs $H$ with large chromatic number. \nThis is joint work with John Haslegrave\, Joseph Hyde and Hong Liu
URL:https://dimag.ibs.re.kr/event/2022-03-21/
LOCATION:Room B232\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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