@prefix vivo: . @prefix edm: . @prefix dcterms: . @prefix dc: . @prefix skos: . @prefix ns0: . vivo:departmentOrSchool "Non UBC"@en ; edm:dataProvider "DSpace"@en ; dcterms:creator "Furedi, Zoltan"@en ; dcterms:issued "2018-08-05T05:00:52Z"@*, "2018-02-05T11:11"@en ; dcterms:description """A {\\em convex geometric hypergraph} or {\\em cgh} is a hypergraph whose vertices form a convex polygon in the plane. It will be convenient to place the vertices in clockwise order on the vertices of a regular polygon $\\vb{\\gamma}_n$ (for $n \\ge 3$), and to write $v_1 < v_2 < \\dots < v_n < v_1$ to denote that the clockwise ordering of the vertices $\\vb{\\gamma}_n$ is $(v_1,v_2,\\dots,v_n,v_1)$. We write cgg (cgh, cg $r$-graph) as abbreviation for convex geometric graph (convex geometric hypergraph, convex geometric $r$-graph), Given an $r$-uniform cgh $F$, let $\\ex_{\\circlearrowright}(n,F)$ be the maximum number of edges in an $r$-uniform cgh on $n$ vertices that does not contain $F$. In the case of graphs, this problem has a rich history with applications to a variety of problems in combinatorial geometry. Extremal questions for convex geometric graphs and hypergraphs are connected to a variety of topics in discrete geometry (see for instance Bra{\\ss}, Rote, and Swanepoel~2001, %\\cite{Brass-Rote-Swanepoel}, Pach and Pinchasi~2013, %\\cite{Pach-Pinchasi}) and questions around the triangle removal problem (see Section 4.3 in Aronov, Dujmovi\\v{c}, Morin, Ooms and da Silveira~2017, %\\cite{Aronov}, Gowers and Long~2016 %\\cite{Gowers-Long} and Loh~2016). %\\cite{Loh}) We focus on the case that $F$ is a certain type of embedding of a tight path or matching. We obtain in many cases asymptotically sharp estimates for $\\ex_{\\circlearrowright}(n,F)$, generalizing classical results of Kupitz and Perles. As a consequence, we show that the number of edges in an $n$-vertex $r$-graph containing no tight $k$-edge path is at most \\[\\frac{(k-1)(r - 1)}{r}{n \\choose r - 1}.\\] The case $r = 2$ is the Erd\\H{o}s-Gallai Theorem. For $r \\geq 3$, this is the first non-trivial upper bound on the extremal number for tight paths in $r$-graphs which applies for all $k \\ge 3$, and marks progress towards Kalai's tight tree conjecture."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/66668?expand=metadata"@en ; dcterms:extent "30 minutes"@en ; dc:format "video/mp4"@en ; skos:note ""@en, "Author affiliation: Renyi Institute of Mathematics"@en ; edm:isShownAt "10.14288/1.0369712"@en ; dcterms:language "eng"@en ; ns0:peerReviewStatus "Unreviewed"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "Banff International Research Station for Mathematical Innovation and Discovery"@en ; dcterms:rights "Attribution-NonCommercial-NoDerivatives 4.0 International"@en ; ns0:rightsURI "http://creativecommons.org/licenses/by-nc-nd/4.0/"@en ; ns0:scholarLevel "Faculty"@en ; dcterms:isPartOf "BIRS Workshop Lecture Videos (Banff, Alta)"@en ; dcterms:subject "Mathematics"@en, "Convex and discrete geometry"@en, "Combinatorics"@en ; dcterms:title "Tight paths and matchings in convex geometric hypergraphs"@en ; dcterms:type "Moving Image"@en ; ns0:identifierURI "http://hdl.handle.net/2429/66668"@en .