图论与组合方向报告

报告人：刘鸿教授

报告时间：2025年4月8日 15:30至18:00

报告地点：中南大学数学与统计学院145教室

标题:Clique density vs blowups

摘要: A well-known theorem of Nikiforov asserts that any graph with a positive $K_r$-density contains a logarithmic blowup of $K_r$. We explore variants of Nikiforov's result and investigate when positive clique density condition implies the existence of a significantly larger blowup of a clique. Our results study such problems for families of ordered graphs with forbidden induced monotone path, obtaining optimal bounds. As corollaries, we strengthen a result of Pach and Tomon, and resolve a conjecture of Tomon in a strong form. To find a large blowup, we reduce the embedding problem to a certain Ramsey problem. For optimal lower bound constructions, we make use of concentration of measure and the isodiametric inequality on high dimensional spheres.

报告人简介：刘鸿，2015年在伊利诺伊大学厄本那-香槟分校（UIUC）取得博士学位，师从József Balogh。2016年在华威大学数学研究所做博士后研究员。后于2019年在华威大学（Warwick）取得终身教职，并摘获英国科研创新未来领袖奖。于2022年加入韩国基础科学研究院（IBS）任首席科学家，现是其极值及概率组合研究组（ECOPRO）的领头人, SIAM Journal on Discrete Mathematics杂志的编委，研究领域包括极值、概率组合、图论、离散几何、组合数论等。刘鸿教授在J. Amer. Math. Soc.，Forum of Mathematics, Pi，American J. Mathematics，J. Euro. Math. Soc.，Proc. London Math. Soc.，J. London Math. Soc.，Proc. Amer. Math. Soc.，Forum of Mathematics, Sigma，J. Combin. Theory Ser. B，J. Combin. Theory Ser. A及Combinatorica等顶级杂志发表多篇学术论文，并多次受邀在国际学术会议上做邀请报告。