报告题目：Difference families: algebraic aspects and extremal aspects

报告人：冯 弢（北京交通大学）

时间：2022年5月20日 星期五 下午3点30分

腾讯会议：777-695-6738 

摘要：Even though Peltesohn proved that a cyclic (v,3,1)-design exists if and only if v ≡ 1,3 (mod 6) as early as 1939, the problem of determining the spectrum of cyclic (v,k,1)-designs with k>3 is far from being settled, even for k=4. By exploring the underlying structure of difference families, we show that a cyclic (v,4,1)-design exists if and only if v ≡ 1,4 (mod 12) and v≠16, 25, 28.

By giving previously unknown a pair of orthogonal orthomorphisms of cyclic groups of order 18t+9 for any positive integer t, we complete the existence spectrum of a pair of orthogonal orthomorphisms of cyclic groups. As a corollary, we complete the existence spectrum of a (G,4,1)-difference matrix over any finite abelian group G. Let D2H be the generalized dihedral group of an abelian group H. It is proved that a (D2H, 4, 1)-difference matrix exists if and only if H is of even order and H is not isomorphic to Z4. It is proved that if G is a finite abelian group and the Sylow 2-subgroup of G is trivial or noncyclic, then a (G,5,1)-DM exists except for some possible exceptions.

Novak conjectured in 1974 that for any cyclic Steiner triple systems of order v with v ≡ 1 (mod 6), it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We generalize this conjecture to cyclic (v, k,λ)-designs with 1≤λ≤k-1. We confirm that the generalization of the conjecture holds when v is a prime andλ=1 by using Combinatorial Nullstellensatz, and also whenλ≤ (k-1)/2 and v is sufficiently large compared to k by using proper edge-coloring of hypergraphs.