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DTSTART:20230101T000000
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DTSTART;TZID=Asia/Seoul:20241213T163000
DTEND;TZID=Asia/Seoul:20241213T173000
DTSTAMP:20260417T234203
CREATED:20241115T050831Z
LAST-MODIFIED:20241116T074247Z
UID:10175-1734107400-1734111000@dimag.ibs.re.kr
SUMMARY:Jun Gao (高峻)\, Phase transition of degenerate Turán problems in p-norms
DESCRIPTION:For a positive real number $p$\, the $p$-norm $\|G\|_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $\mathrm{ex}_{p}(n\,F)$ of $F$-free graphs on $n$ vertices\, focusing on the case where $F$ is a bipartite graph. It is natural to conjecture that for every bipartite graph $F$\, there exists a threshold $p_F$ such that for $p< p_{F}$\, the order of $\mathrm{ex}_{p}(n\,F)$ is governed by pseudorandom constructions\, while for $p > p_{F}$\, it is governed by star-like constructions. We determine the exact value of $p_{F}$\, under a mild assumption on the growth rate of $\mathrm{ex}(n\,F)$. Our results extend to $r$-uniform hypergraphs as well. \nWe also prove a general upper bound that is tight up to a $\log n$ factor for $\mathrm{ex}_{p}(n\,F)$ when $p = p_{F}$.\nWe conjecture that this $\log n$ factor is unnecessary and prove this conjecture for several classes of well-studied bipartite graphs\, including one-side degree-bounded graphs and families of short even cycles. \nThis is a joint work with Xizhi Liu\, Jie Ma and Oleg Pikhurko.
URL:https://dimag.ibs.re.kr/event/2024-12-13/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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