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SUMMARY:Hong Liu (University of Warwick)
DTSTART:20210208T140000Z
DTEND:20210208T150000Z
DTSTAMP:20260423T035530Z
UID:EPC/46
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EPC/46/">Opt
 imal high dimension geometric construction for Ramsey-Turan theory</a>\nby
  Hong Liu (University of Warwick) as part of Extremal and probabilistic co
 mbinatorics webinar\n\n\nAbstract\nCombining two classical notions in extr
 emal graph theory\, the study of Ramsey-Turan theory seeks to determine\, 
 for integers $m\\le n$ and $p \\leq q$\, the number $RT_p(n\,K_q\,m)$\, wh
 ich is the maximum size of an $n$-vertex $K_q$-free graph in which every s
 et of at least $m$ vertices contains a $K_p$.\n\nTwo major open problems i
 n this area from the 80s ask:\n(1) whether the asymptotic extremal structu
 re for the general case exhibits certain periodic behaviour\, resembling t
 hat of the special case when $p=2$ \;\n(2) to construct analogues of the B
 ollobas-Erdos graph with densities other than powers of $1/2$.\n\nWe refut
 e the first conjecture by witnessing asymptotic extremal structures that a
 re drastically different from the $p=2$ case\; and address the second prob
 lem by constructing Bollobas-Erdos-type graphs with any rational density u
 p to $\\frac{1}{2}$ using high dimension complex sphere.\n\nJoint work wit
 h Christian Reiher\, Maryam Sharifzadeh and Katherine Staden.\n
LOCATION:https://researchseminars.org/talk/EPC/46/
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