A Note on t-designs with t Intersection Numbers

Let X be a finite set of v elements, called points, and let β be a finite family of distinct k-subsets of X , called blocks. Then the pair D = (X ,β) is called a t-design with parameters (v,k,λ) if any t-subset of X is contained in exactly λ members of β. If λi denotes the number of blocks containing i points, i = 0,1,2, . . . , t−1, then λi is independent of the choice of the i points and λi (k−i t−i ) = λ (v−i t−i ) . In particular, b = λ0 is the number of blocks, and λ1 = r is the number of blocks through any point of D. A 0-design is a pair (X ,β) where β is a collection of k-subsets of X . For 0 ≤ x < k, x is called intersection number of D if there exists B,B′ ∈ β such that |B∩B′| = x. A 2-design with two intersection numbers is said to be a quasi-symmetric design. For a 0-design with t intersection numbers, Ray-Chaudhuri and Wilson proved that b ≤ (v t ) (see [1]). This result is used by Sane and Shrikhande [7] to prove that, for a fixed value of the block size k, there exist finitely many quasi-symmetric designs with λ > 1. In the literature many finiteness results for quasi-symmetric designs and quasi-symmetric 3-designs are proved by using this result by Sane and Shrikhande (see [5], [6], [7], [8]). Our main aim in this paper is to extend the result by Sane and Shrikhande to t-designs with t intersection numbers. More specifically, we obtain a relation between the parameters of a t-design with t intersection numbers, and we use it to show that, for a fixed value of the block size k, v takes finitely many values. Finally, we use the result by Ray-Chaudhuri and Wilson to complete the proof.