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SUMMARY:Ethan White (University of British Columbia)
DTSTART:20200604T193000Z
DTEND:20200604T195500Z
DTSTAMP:20260423T011224Z
UID:CANT2020/54
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2020/54/
 ">Directions in $AG(2\,p)$ and the clique number of Paley graphs</a>\nby E
 than White (University of British Columbia) as part of Combinatorial and a
 dditive number theory (CANT 2021)\n\n\nAbstract\nThe directions determined
  by a point set are the slopes of lines passing through at least two \npoi
 nts of the set. A seminal result of Rédei tells us that at least $(p+3)/2
 $ directions are determined \nby $p$ points in $AG(2\,p)$. We consider car
 tesian product point sets\, i.e. a set of the form \n$A \\times B \\subset
  AG(2\,p)$\, where $p$ is prime\, $A$ and $B$ are subsets of $GF(p)$ each 
 \nwith at least two elements and $|A||B| <p$. In this case\, we show that 
 the number of directions \ndetermined is at least $|A||B| - \\min\\{|A|\,|
 B|\\} + 2$.  This gives an upper bound of about $\\sqrt{p/2}$ \non the cli
 que number of Paley graphs\, matching a bound obtained by Hanson and Petri
 dis last year. \nOur main tool is the use of the R\\'edei polynomial with 
 Sz\\H{o}nyi's extension. \n\nJoint work with Józseff Solymosi and Daniel 
 Di Benedetto.\n
LOCATION:https://researchseminars.org/talk/CANT2020/54/
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