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:20260423T021041Z UID:EPC/3 DESCRIPTION:Title: Tour nament quasirandomness from local counting\nby Matija Bucić (ETH Zuri ch) as part of Extremal and probabilistic combinatorics webinar\n\n\nAbstr act\nA well-known theorem of Chung and Graham states that if h>3 then a to urnament T is quasirandom if and only if T contains each h-vertex tourname nt the "correct number" of times as a subtournament. In this talk we inves tigate the relationship between quasirandomness of T and the count of a si ngle h-vertex tournament H in T. We consider two types of counts\, the glo bal one and the local one.\n\nWe first observe that if T has the correct g lobal count of H and h>6 then quasirandomness of T is only forced if H is transitive. The next natural question when studying quasirandom objects as ks whether possessing the correct local counts of H is enough to force qua sirandomness 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 studied before in both the graph and hypergraph settings. Perhaps the closest anal ogue of our problem was considered by Simonovits and Sós who looked at wh ether having "correct counts" of a fixed graph H as an induced subgraph of G implies G must be quasirandom\, in an appropriate sense. They proved th at this is indeed the case when H is regular and conjectured that it holds for all H (except the path on 3 vertices).\n\nContrary to the Simonovits- Sós conjecture\, in the tournament setting we prove that a constant propo rtion of all tournaments are not locally forcing. In fact\, any locally fo rcing tournament must itself be strongly quasirandom. On the other hand\, unlike the global forcing case\, we construct infinite families of non-tra nsitive locally forcing tournaments.\n LOCATION:https://researchseminars.org/talk/EPC/3/ END:VEVENT END:VCALENDAR