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Let K = { k 1 , k 2 , … , k r } and L = { l 1 , l 2 , … , l s } be disjoint subsets of { 0 , 1 , … , p − 1 } , where p is a prime and A = { A 1 , A 2 , … , A m } is a family of subsets of [ n ] = { 1 , 2 , … , n } such that | A i | ( mod p ) ∈ K for all A i ∈ A and | A i ∩ A j | ( mod p ) ∈ L for i ≠ j . In 1991, Alon, Babai and Suzuki conjectured that if n ≥ s + max 1 ≤ i ≤ r k i , then | A | ≤ n s + n s − 1 + ⋯ + n s − r + 1 . In 2000, Qian and Ray-Chaudhuri proved the conjecture under the condition n ≥ 2 s − r . In 2015, Hwang and Kim verified this conjecture. In this paper, we will prove that if n ≥ 2 s − 2 r + 1 or n ≥ s + max 1 ≤ i ≤ r k i , then | A | ≤ n − 1 s + n − 1 s − 1 + ⋯ + n − 1 s − 2 r + 1 . This result strengthens both the upper bound of Alon–Babai–Suzuki’s conjecture and Qian and Ray-Chaudhuri’s result, when n ≥ 2 s − 2 . [ABSTRACT FROM AUTHOR]

Abstract: Non-uniform group divisible designs (GDDs) and non-uniform Kirkman frames are useful in the constructions for other types of designs. In this paper, we consider the existence problems for -GDDs of type with and Kirkman frames of type . First, we determine completely the spectrum for uniform -GDDs of type . Then, we consider the entire existence problem for non-uniform -GDDs of type with . We show that, for each given , up to a small number of undetermined cases of , the necessary conditions on for the existence of a -GDD of type are also sufficient, except possibly when for and when for . Finally, a similar result for Kirkman frames of type is obtained. We show that, for each given , up to a small number of undetermined cases of , the necessary conditions on for the existence of a Kirkman frame of type are also sufficient, except possibly when for and when for . [Copyright &y& Elsevier]

Abstract: Non-uniform group divisible designs are instrumental in the constructions for other types of designs. Most of the progress for the existence of -GDDs of type is on the case when is even, where the existence for small has played a key role. In order to determine the spectrum for -GDDs of type with being odd, we continue to investigate the small cases with in this paper. We show that, for each , the necessary conditions for the existence of a -GDD of type are also sufficient. As the applications of these GDDs, we obtain a few pairwise balanced designs with minimum block size 4. Meanwhile, we also improve the existence result for frame self-orthogonal Mendelsohn triple systems of type by reducing an infinite class of possible exceptions, namely and , to eight undetermined cases. [Copyright &y& Elsevier]