A finiteness theorem of block-transitive point-imprimitive 3-designs

作者：Zhu, Yunfei; Guan, Haiyan^*

来源：Discrete Mathematics, 2024, 347(3): 113819.

DOI：10.1016/j.disc.2023.113819

• 3-design
• Block-transitive
• Point-imprimitive

In 2003, Tuan showed a finiteness theorem for block-transitive point-imprimitive 3-(k(k-1) / 2 + 1, k, lambda) designs. As a generalization of this result, we consider the block-transitive pointimprimitive 3-(v, k, lambda) designs with v < k(k-1) / 2 + 1. Let D = (P, B) be a 3-(v, k, lambda) design admitting G as a block-transitive point-imprimitive automorphism group, and G preserve a partition C of the points into d imprimitivity classes of size c, here v = wk(k-1) / s + 1, s, w are two positive integers with gcd(s, w) =1 and 2w < s. We prove that, for a given positive integer s, there are only finitely many numbers v such that there exist nontrivial block-transitive point-imprimitive 3-(v, k, lambda) designs with c, d >= 3. Moreover, we obtain the classification for this type of designs when s <= 10.