Title: Exceptions and characterization results for type‐1 λ‐designs
Abstract: Let X be a finite set with v elements, called points and β be a family of subsets of X, called blocks. A pair ( X , β ) is called λ-design whenever ∣ β ∣ = ∣ X ∣ and 1. for all B i , B j ∈ β , i ≠ j , ∣ B i ∩ B j ∣ = λ; 2. for all B j ∈ β , ∣ B j ∣ = k j > λ, and not all k j are equal. The only known examples of λ-designs are so-called type-1 designs, which are obtained from symmetric designs by a certain complementation procedure. Ryser and Woodall had independently conjectured that all λ-designs are type-1. Let r , r * ( r > r * ) be replication numbers of a λ-design D = ( X , β ) and g = gcd ( r − 1 , r * − 1 ) , m = gcd ( ( r − r * ) ∕ g , λ ) , and m ′ = m, if m is odd and m ′ = m ∕ 2, otherwise. For distinct points x and y of D, let λ ( x , y ) denote the number of blocks of X containing x and y. We strengthen a lemma of S.S. Shrikhande and N.M. Singhi and use it to prove that if r ( r − 1 ) ( v − 1 ) − k ( r − r * ) m ′ ( v − 1 ) are not integers for k = 1 , 2 , … , m ′ − 1, then D is type-1. As an application of these results, we show that for fixed positive integer θ there are finitely many nontype-1 λ-designs with r = r * + θ. If r − r * = 27 or r − r * = 4 p and r * ≠ ( p − 1 ) 2 , or v = 7 p + 1 such that p ≢ 1 , 13 ( mod 21 ) and p ≢ 4 , 9 , 19 , 24 ( mod 35 ) , where p is a positive prime, then D is type-1. We further obtain several inequalities involving λ ( x , y ) , where equality holds if and only if D is type-1.
Publication Year: 2020
Publication Date: 2020-05-18
Language: en
Type: article
Indexed In: ['crossref']
Access and Citation
Cited By Count: 2
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