Exceptions and characterization results for type‐1 λ ‐designs

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 ifr ( 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.