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SUMMARY:Alex Rice (Millsaps College)
DTSTART:20220526T193000Z
DTEND:20220526T195500Z
DTSTAMP:20260423T011340Z
UID:CANT2022/42
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2022/42/
 ">New results in classical and arithmetic Ramsey theory</a>\nby Alex Rice 
 (Millsaps College) as part of Combinatorial and additive number theory (CA
 NT 2022)\n\n\nAbstract\nFor $r\,k\\in \n$\, Ramsey's Theorem says that the
 re exists a least positive integer $R_r(k)$ such that every $r$-coloring o
 f the edges of a complete graph on $N\\geq R_r(k)$ vertices yields a monoc
 hromatic complete subgraph on $k$ vertices. This fact can be applied to de
 duce Schur's Theorem\, which says that there exists a least positive integ
 er $S_r(k)$ such that every $r$-coloring of $\\{1\,2\,\\dots\,N\\}$ for $N
 \\geq S_r(k)$ yields a monochromatic solution to the equation $x_1+x_2+\\c
 dots+x_{k-1}=x_k$. Here we discuss new findings related to these two class
 ical results. First\, we derive explicit upper bounds on $R_r(k)$\, establ
 ished through the pigeonhole principle and careful bookkeeping\, that impr
 ove upon previously documented bounds. Second\, we present an extension of
  Schur's Theorem to higher-dimensional integer lattices\, with the additio
 nal restriction that the vectors on the left hand side of the equation are
  linearly independent. \n\nThis includes joint work with six (at the time)
  Millsaps College undergraduate students: Vishal Balaji\, Powers Lamb\, An
 drew Lott\, Dhruv Patel\, Sakshi Singh\, and Christine Rose Ward.\n
LOCATION:https://researchseminars.org/talk/CANT2022/42/
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