BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Natasha Morrison (Victoria) DTSTART:20211117T160000Z DTEND:20211117T170000Z DTSTAMP:20260423T003235Z UID:WCS/38 DESCRIPTION:Title: Unc ommon systems of equations\nby Natasha Morrison (Victoria) as part of Warwick Combinatorics Seminar\n\n\nAbstract\nA system of linear equations $L$ over $\\mathbb{F}_q$ is common if the number of monochromatic solution s to $L$ in any two-colouring of $\\mathbb{F}_q^n$ is asymptotically at le ast the expected number of monochromatic solutions in a random two-colouri ng of $\\mathbb{F}_q^n$. Motivated by existing results for specific system s (such as Schur triples and arithmetic progressions)\, as well as extensi ve research on common and Sidorenko graphs\, the systematic study of commo n systems of linear equations was recently initiated by Saad and Wolf. Bui lding on earlier work of of Cameron\, Cilleruelo and Serra\, as well as Sa ad and Wolf\, common linear equations have been fully characterised by Fox \, Pham and Zhao.\n\nIn this talk I will discuss some recent progress towa rds a characterisation of common systems of two or more equations. In part icular we prove that any system containing an arithmetic progression of le ngth four is uncommon\, confirming a conjecture of Saad and Wolf. This fol lows from a more general result which allows us to deduce the uncommonness of a general system from certain properties of one- or two-equation subsy stems.\n LOCATION:https://researchseminars.org/talk/WCS/38/ END:VEVENT END:VCALENDAR