BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Nina Kamcev (University of Zagreb) DTSTART:20211125T151500Z DTEND:20211125T170000Z DTSTAMP:20260422T065852Z UID:CJCS/44 DESCRIPTION:Title: Co mmon systems of equations are rare\nby Nina Kamcev (University of Zagr eb) as part of Copenhagen-Jerusalem Combinatorics Seminar\n\n\nAbstract\nS everal classical results in Ramsey theory (including famous theorems of Sc hur\, van der Waerden\, Rado) deal with finding monochromatic linear patte rns in two-colourings of the integers. Our topic will be quantitative exte nsions of such results.\n \nA linear system $L$ over $\\mathcal{F}_q$ is \ \emph{common} if the number of monochromatic solutions to $L=0$ in any two -colouring of $\\mathcal{F}_q^n$ is asymptotically at least the expected n umber of monochromatic solutions in a random two-colouring of $\\mathcal{F }_q^n$. Motivated by existing results for specific systems (such as Schur triples and arithmetic progressions)\, as well as extensive research on co mmon and Sidorenko graphs\, the systematic study of common systems of line ar equations was recently initiated by Saad and Wolf. Fox\, Pham and Zhao characterised common linear equations. \n \nI will talk about recent prog ress towards a classification of common systems of two or more linear equa tions. In particular\, any system containing a four-term arithmetic progre ssion is uncommon. 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 \nJoint work with Anita Liebenau and N atasha Morrison.\n LOCATION:https://researchseminars.org/talk/CJCS/44/ END:VEVENT END:VCALENDAR