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DTSTART:20220101T000000
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DTSTART;TZID=Asia/Seoul:20231128T163000
DTEND;TZID=Asia/Seoul:20231128T173000
DTSTAMP:20260418T193439
CREATED:20231031T121453Z
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UID:7831-1701189000-1701192600@dimag.ibs.re.kr
SUMMARY:Hyunwoo Lee (이현우)\, Towards a high-dimensional Dirac's theorem
DESCRIPTION:Dirac’s theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering hypergraph matchings and Hamiltonian cycles. \nWe consider another natural generalization of the perfect matchings\, Steiner triple systems. As a Steiner triple system can be viewed as a partition of pairs of vertices\, it is a natural high-dimensional analogue of a perfect matching in graphs. \nWe prove that for sufficiently large integer $n$ with $n \equiv 1 \text{ or } 3 \pmod{6}\,$ any $n$-vertex $3$-uniform hypergraph $H$ with minimum codegree at least $\left(\frac{3 + \sqrt{57}}{12} + o(1) \right)n = (0.879… + o(1))n$ contains a Steiner triple system. In fact\, we prove a stronger statement by considering transversal Steiner triple systems in a collection of hypergraphs. \nWe conjecture that the number $\frac{3 + \sqrt{57}}{12}$ can be replaced with $\frac{3}{4}$ which would provide an asymptotically tight high-dimensional generalization of Dirac’s theorem.
URL:https://dimag.ibs.re.kr/event/2023-11-28/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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