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SUMMARY:Anush Tserunyan (McGill University)
DTSTART:20201119T190000Z
DTEND:20201119T200000Z
DTSTAMP:20260423T021311Z
UID:OLS/25
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OLS/25/">Con
 tainers made easy</a>\nby Anush Tserunyan (McGill University) as part of O
 nline logic seminar\n\n\nAbstract\nA modern trend in extremal combinatoric
 s is extending classical results from the dense setting (e.g. Szemer&eacut
 e\;di's theorem) to the sparse random setting. More precisely\, one shows 
 that a property of a given ``dense'' structure is inherited by a randomly 
 chosen ``sparse'' substructure. A recent breakthrough tool for proving suc
 h statements is the Balogh--Morris--Samotij and Saxton--Thomason hypergrap
 h containers method\, which bounds the number of independent sets in homog
 eneously dense finite hypergraphs\, thus implying that a random sparse sub
 set is not independent. In a joint work with A. Bernshteyn\, M. Delcourt\,
  and H. Towsner\, we give a new --- elementary and nonalgorithmic --- proo
 f of the containers theorem for finite hypergraphs. Our proof is inspired 
 by considering hyperfinite hypergraphs in the setting of nonstandard analy
 sis\, where there is a notion of dimension capturing the logarithmic rate 
 of growth of finite sets. Applying this intuition in another setting with 
 a notion of dimension\, namely\, algebraically closed fields\, A. Bernshte
 yn\, M. Delcourt\, and I prove an analogous theorem for ``dense'' algebrai
 cally definable hypergraphs: any Zariski-generic low-dimensional subset of
  such hypergraphs is itself ``dense'' (in particular\, not independent).\n
LOCATION:https://researchseminars.org/talk/OLS/25/
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