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SUMMARY:Alexander Clifton (Emory University)
DTSTART:20220217T151500Z
DTEND:20220217T160000Z
DTSTAMP:20260422T065545Z
UID:CJCS/55
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CJCS/55/">Ra
 msey Theory for Diffsequences</a>\nby Alexander Clifton (Emory University)
  as part of Copenhagen-Jerusalem Combinatorics Seminar\n\n\nAbstract\nVan 
 der Waerden's theorem states that any coloring of $\\mathbb{N}$ with a fin
 ite number of colors will contain arbitrarily long monochromatic arithmeti
 c progressions. This motivates the definition of the van der Waerden numbe
 r $W(r\,k)$ which is the smallest $n$ such that any $r$-coloring of $\\{1\
 ,2\,\\cdots\,n\\}$ guarantees the presence of a monochromatic arithmetic p
 rogression of length $k$.\n\nIt is natural to ask what other arithmetic st
 ructures exhibit van der Waerden-type results. One notion\, introduced by 
 Landman and Robertson\, is that of a $D$-diffsequence\, which is an increa
 sing sequence $a_1 < a_2 <\\cdots< a_k$ in which the consecutive differenc
 es $a_i-a_{i-1}$ all lie in some given set $D$. For each $D$ that exhibits
  a van der Waerden-type result\, we let $\\Delta(D\,k\;r)$ represent the a
 nalogue of the van der Waerden number $W(r\,k)$. One question of interest 
 is to determine the possible behaviors of $\\Delta$ as a function of $k$. 
 In this talk\, we will demonstrate that it is possible for $\\Delta(D\,k\;
 r)$ to grow faster than polynomial in $k$. Time permitting\, we will also 
 discuss a class of $D$'s for which no van der Waerden-type result is possi
 ble.\n
LOCATION:https://researchseminars.org/talk/CJCS/55/
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