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SUMMARY:Collier Gaiser (Community College of Aurora\, Colorado)
DTSTART:20260718T203000Z
DTEND:20260718T205500Z
DTSTAMP:20260710T111502Z
UID:CANT2026/92
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/92/
 ">On Rado numbers for equations with unit fractions</a>\nby Collier Gaiser
  (Community College of Aurora\, Colorado) as part of Combinatorial and add
 itive number theory seminar (CANT 2026)\n\nLecture held in Science Center 
 in the CUNY Graduate Center (4th floor).\n\nAbstract\nLet $R_r(k)$ be the 
 smallest $n$ such that every $r$-coloring of $\\{1\,2\,...\,n\\}$ has a mo
 nochromatic solution to $x_1+x_2+\\cdots+x_k=y$\, where $x_1\,x_2\,\\ldots
 \,x_k$ are not necessarily distinct. Beutelspacher and Brestovansky proved
  that $R_2(k)=k^2+k-1$ and\, recently\, Boza\, Mar\\'{i}n\, Revuelta\, and
  Sanz proved that $R_3(k)=k^3+2k^2-2$. Similarly\, let $f_r(k)$ be the sma
 llest $n$ such that every $r$-coloring of $\\{1\,2\,...\,n\\}$ has a monoc
 hromatic solution to the equation $1/x_1+1/x_2+\\cdots+1/x_k=1/y$\, where 
 $x_1\,x_2\,\\ldots\,x_k$ are not necessarily distinct. Brown and R\\"{o}dl
  proved that $f_2(k)=O(k^6)$. In this talk\, we show that $f_2(k)=O(k^3)$ 
 and $f_3(k)=O(k^{43})$. The main ingredient in our proof is a finite set $
 A\\subseteq\\mathbb{N}$ such that every $r$-coloring of $A$ has a monochro
 matic solution to the linear equation $x_1+x_2+\\cdots+x_k=y$ and the leas
 t common multiple of $A$ is sufficiently small. As for the lower bound\, w
 e show that $f_r(k)\\geq k^r$ which leads to an interesting open question:
  Is $f_2(k)=\\Theta(k^2)$?\n
LOCATION:https://researchseminars.org/talk/CANT2026/92/
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