»ã±¨±êÌâ (Title)£ºGeneralized Kleitman¡¯s theorem

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»ã±¨ÈË (Speaker)£ºÅ·ÑôÃú»Ô ²©Ê¿£¨±±¾©´óѧ£©

»ã±¨¹¦·ò (Time)£º2025Äê01ÔÂ08ÈÕ(ÖÜÈý) 10:00

»ã±¨µØÖ· (Place)£ºÐ£±¾²¿GJ203

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»ã±¨ÌáÒª£ºGiven 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 Cvetkovi? 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.