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SUMMARY:The number of monochromatic solutions to multiplicative equations 
 - Victor Souza (Cambridge)
DTSTART:20241024T133000Z
DTEND:20241024T143000Z
UID:TALK223576@talks.cam.ac.uk
CONTACT:103978
DESCRIPTION:Given an r-colouring of the interval {2\,...\,N}\, what is the
  minimum number of monochromatic solutions of the equation xy = z? For r =
  2\, we show that there are always asymptotically at least (1/2sqrt(2)) N^
 (1/2) log(N) monochromatic solutions\, and that the leading constant is sh
 arp. We also establish a stability version of this result. For general r\,
  we show that there are at least C_r N^(1/S(r-1)) monochromatic solutions\
 , where S(r) is the Schur number for r colours and C_r is a constant. This
  bound is sharp up to logarithmic factors when r <= 4. We also obtain resu
 lts for more general multiplicative equations.\n
LOCATION:MR12
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