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DTSTART;TZID=America/New_York:20220503T093000
DTEND;TZID=America/New_York:20220503T103000
DTSTAMP:20260527T171024
CREATED:20240214T072025Z
LAST-MODIFIED:20240304T055241Z
UID:10002557-1651570200-1651573800@cmsa.fas.harvard.edu
SUMMARY:The threshold for stacked triangulations
DESCRIPTION:Abstract: Consider a bootstrap percolation process that starts with a set of `infected’ triangles $Y \subseteq \binom{[n]}3$\, and a new triangle f gets infected if there is a copy of K_4^3 (= the boundary of a tetrahedron) in which f is the only not-yet infected triangle.\nSuppose that every triangle is initially infected independently with probability p=p(n)\, what is the threshold probability for percolation — the event that all triangles get infected? How many new triangles do get infected in the subcritical regime? \nThis notion of percolation can be viewed as a simplification of simple-connectivity. Namely\, a stacked triangulation of a triangle is obtained by repeatedly subdividing an inner face into three faces.\nWe ask: for which $p$ does the random simplicial complex Y_2(n\,p) contain\, for every triple $xyz$\, the faces of a stacked triangulation of $xyz$ whose internal vertices are arbitrarily labeled in [n]. \nWe consider this problem in every dimension d>=2\, and our main result identifies a sharp probability threshold for percolation\, showing it is asymptotically (c_d*n)^(-1/d)\, where c_d is the growth rate of the Fuss–Catalan numbers of order d. \nThe proof hinges on a second moment argument in the supercritical regime\, and on Kalai’s algebraic shifting in the subcritical regime. \nJoint work with Eyal Lubetzky.
URL:https://cmsa.fas.harvard.edu/event/5-3-2022-cmsa-combinatorics-physics-and-probability-seminar/
LOCATION:Hybrid
CATEGORIES:Combinatorics Physics and Probability
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Combinatorics-Physics-and-Probability-Seminar-05.03.22-1.png
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DTSTART;TZID=America/New_York:20220517T093000
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CREATED:20240214T072604Z
LAST-MODIFIED:20240304T055019Z
UID:10002559-1652779800-1652783400@cmsa.fas.harvard.edu
SUMMARY:Hypergraph Matchings Avoiding Forbidden Submatchings
DESCRIPTION:Abstract:  In 1973\, Erdős conjectured the existence of high girth (n\,3\,2)-Steiner systems. Recently\, Glock\, Kühn\, Lo\, and Osthus and independently Bohman and Warnke proved the approximate version of Erdős’ conjecture. Just this year\, Kwan\, Sah\, Sawhney\, and Simkin proved Erdős’ conjecture. As for Steiner systems with more general parameters\, Glock\, Kühn\, Lo\, and Osthus conjectured the existence of high girth (n\,q\,r)-Steiner systems. We prove the approximate version of their conjecture.  This result follows from our general main results which concern finding perfect or almost perfect matchings in a hypergraph G avoiding a given set of submatchings (which we view as a hypergraph H where V(H)=E(G)). Our first main result is a common generalization of the classical theorems of Pippenger (for finding an almost perfect matching) and Ajtai\, Komlós\, Pintz\, Spencer\, and Szemerédi (for finding an independent set in girth five hypergraphs). More generally\, we prove this for coloring and even list coloring\, and also generalize this further to when H is a hypergraph with small codegrees (for which high girth designs is a specific instance). Indeed\, the coloring version of our result even yields an almost partition of K_n^r into approximate high girth (n\,q\,r)-Steiner systems.  If time permits\, I will explain some of the other applications of our main results such as to rainbow matchings.  This is joint work with Luke Postle.
URL:https://cmsa.fas.harvard.edu/event/5-17-2022-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Combinatorics-Physics-and-Probability-Seminar-05.17.22.png
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