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SUMMARY:Natasha Morrison (University of Victoria)
DTSTART:20211101T140000Z
DTEND:20211101T150000Z
DTSTAMP:20260423T035537Z
UID:EPC/75
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EPC/75/">Unc
 ommon systems of equations</a>\nby Natasha Morrison (University of Victori
 a) as part of Extremal and probabilistic combinatorics webinar\n\n\nAbstra
 ct\nA system of linear equations $L$ over $\\mathbb{F}_q$ is common if the
  number of monochromatic solutions to $L$ in any two-colouring of $\\mathb
 b{F}_q^n$ is asymptotically at least the expected number of monochromatic 
 solutions in a random two-colouring of $\\mathbb{F}_q^n$. Motivated by exi
 sting results for specific systems (such as Schur triples and arithmetic p
 rogressions)\, as well as extensive research on common and Sidorenko graph
 s\, the systematic study of common systems of linear equations was recentl
 y initiated by Saad and Wolf. Building on earlier work of Cameron\, Ciller
 uelo and Serra\, as well as Saad and Wolf\, common linear equations have b
 een fully characterised by Fox\, Pham and Zhao.\n\nIn this talk I will dis
 cuss some recent progress towards a characterisation of common systems of 
 two or more equations. In particular we prove that any system containing a
 n arithmetic progression of length four is uncommon\, confirming a conject
 ure of Saad and Wolf. This follows from a more general result which allows
  us to deduce the uncommonness of a general system from certain properties
  of one- or two-equation subsystems.\n
LOCATION:https://researchseminars.org/talk/EPC/75/
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