BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Matija Bucić (ETH Zurich) DTSTART:20200504T130000Z DTEND:20200504T140000Z DTSTAMP:20260404T163658Z UID:EPC/3 DESCRIPTION:Title: Tournament quasirandomness from local counting\nby Matija Bucić (E TH Zurich) as part of Extremal and probabilistic combinatorics webinar\n\n \nAbstract\nA well-known theorem of Chung and Graham states that if h>3 th en a tournament T is quasirandom if and only if T contains each h-vertex t ournament the "correct number" of times as a subtournament. In this talk w e investigate the relationship between quasirandomness of T and the count of a single h-vertex tournament H in T. We consider two types of counts\, the global one and the local one.\n\nWe first observe that if T has the co rrect global count of H and h>6 then quasirandomness of T is only forced i f H is transitive. The next natural question when studying quasirandom obj ects asks whether possessing the correct local counts of H is enough to fo rce quasirandomness of T. A tournament H is said to be locally forcing if it has this property.\n\nVariants of the local forcing problem have been s tudied before in both the graph and hypergraph settings. Perhaps the close st analogue of our problem was considered by Simonovits and Sós who looke d at whether having "correct counts" of a fixed graph H as an induced subg raph of G implies G must be quasirandom\, in an appropriate sense. They pr oved that this is indeed the case when H is regular and conjectured that i t holds for all H (except the path on 3 vertices).\n\nContrary to the Simo novits-Sós conjecture\, in the tournament setting we prove that a constan t proportion of all tournaments are not locally forcing. In fact\, any loc ally forcing tournament must itself be strongly quasirandom. On the other hand\, unlike the global forcing case\, we construct infinite families of non-transitive locally forcing tournaments.\n LOCATION:https://stable.researchseminars.org/talk/EPC/3/ END:VEVENT END:VCALENDAR