报告题目 (Title)：Generalized Kleitman’s theorem
中文标题：广义Kleitma定理
报告人 (Speaker)：欧阳铭晖（北京大学）
报告时间 (Time)：2025年01月08日(周三) 10:00
报告地点 (Place)：校本部GJ203
邀请人(Inviter)：冷岗松、席东盟、李晋、吴加勇
主办部门：理学院数学系
报告摘要：Given distance set D, what is the maximum volume of a subset of {0,1}^n such that the Hamming distance between any distinct vertices belongs to D? When D = {1,2,...,k}, Kleitman established a result stating that the maximum volume of D-distance family is attained by the union of one or two adjacent radius-(k/2) Hamming ball depending on whether k is even or odd. Huang, Klurman, and Pohoata gave a new algebraic proof of Kleitman’s theorem based on the Cvetković bound on independence numbers, and investigated the case when D = {2s+1,...,2t}. They conjectured that the near-perfect (n,t,t-s)-design asymptotically attains the maximum volume among D-distance families. We generalize Huang, Klurman, and Pohoata's method giving an exact result in the case D = {2,4,...,2s}, and get an asymptotically tight result for the maximum volume of D-distance family on {0,1}^n for any homogeneous arithmetic progression D = {sd,(s+1)d,...,td}. This confirms Huang-Klurman-Pohoata's conjecture. Joint work with Zichao Dong, Jun Gao, Hong Liu, and Qiang Zhou.
