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DTSTART;TZID=Asia/Seoul:20220927T163000
DTEND;TZID=Asia/Seoul:20220927T173000
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UID:6074-1664296200-1664299800@dimag.ibs.re.kr
SUMMARY:Alexander Clifton\, Ramsey Theory for Diffsequences
DESCRIPTION:Van der Waerden’s theorem states that any coloring of $\mathbb{N}$ with a finite number of colors will contain arbitrarily long monochromatic arithmetic progressions. This motivates the definition of the van der Waerden number $W(r\,k)$ which is the smallest $n$ such that any $r$-coloring of $\{1\,2\,\cdots\,n\}$ guarantees the presence of a monochromatic arithmetic progression of length $k$. \nIt is natural to ask what other arithmetic structures exhibit van der Waerden-type results. One notion\, introduced by Landman and Robertson\, is that of a $D$-diffsequence\, which is an increasing sequence $a_1<a_2<\cdots<a_k$ in which the consecutive differences $a_i-a_{i-1}$ all lie in some given set $D$. We say that $D$ is $r$-accessible if every $r$-coloring of $\mathbb{N}$ contains arbitrarily long monochromatic $D$-diffsequences. When $D$ is $r$-accessible\, we define $\Delta(D\,k;r)$ as the smallest $n$ such that any $r$-coloring of $\{1\,2\,\cdots\,n\}$ guarantees the presence of a monochromatic $D$-diffsequence of length $k$. \nOne question of interest is to determine the possible behaviors of $\Delta$ as a function of $k$. In this talk\, we will demonstrate that is possible for $\Delta(D\,k;r)$ to grow faster than polynomial in $k$. We will also discuss a broad class of $D$’s which are not $2$-accessible.
URL:https://dimag.ibs.re.kr/event/2022-09-27/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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