BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Hao Huang (Emory University) DTSTART:20201026T140000Z DTEND:20201026T150000Z DTSTAMP:20260423T035531Z UID:EPC/28 DESCRIPTION:Title: On local Turán problems\nby Hao Huang (Emory University) as part of Extr emal and probabilistic combinatorics webinar\n\n\nAbstract\nSince its form ulation\, Turán's hypergraph problems have been among the most challengin g open problems in extremal combinatorics. One of them is the following: g iven a 3-uniform hypergraph F on n vertices in which any five vertices spa n at least one edge\, prove that $|F|\\ge(1/4 -o(1))\\binom{n}{3}$. The co nstruction showing that this bound would be best possible is simply ${X \\ choose 3} \\cup {Y \\choose 3}$ where X and Y evenly partition the vertex set. This construction satisfies the following more general (2p+1\, p+1)- property: any set of 2p+1 vertices spans a complete sub-hypergraph on p+1 vertices.\n\nIn this talk\, we will show that\, quite surprisingly\, for a ll p>2 the (2p+1\,p+1)-property implies the conjectured lower bound. Furth ermore\, we will prove that for integers r\, a >= 2\, the minimum edge den sity of an r-uniform hypergraph satisfying the (ap+1\, p+1)-property tends to $1/a^{r-1}$ when p tends to infinity.\n\nJoint work with Peter Frankl and Vojtěch Rödl.\n LOCATION:https://researchseminars.org/talk/EPC/28/ END:VEVENT END:VCALENDAR