An Elementary Proof of the Intersection Number Property

A 2 − (υ, k,λ ) design (also called a balanced incomplete block design) is an incidence structure consisting of a set P of υ points and a collection B of b blocks, each block being a k-subset of P , such that every point appears in exactly r blocks and every unordered pair of distinct points appears together in exactly λ blocks. The parameters satisfy the fundamental relations

Fisher’s inequality b ≥υ is one of the earliest and most important results in design theory. When b =υ (hence r = k), the design is called symmetric; when b >υ, it is called nonsymmetric. Symmetric designs possess many remarkable properties; for instance, any two blocks in a symmetric design intersect in exactly λ points. This classical result has been proved in various ways, including incidence matrix methods and combinatorial arguments. For further standard notation and definitions, the reader is referred to sources such as [1,2].

The study of block intersection numbers—the sizes of intersections between distinct blocks—has been a fruitful area in design theory. For a 2-design, let x₁,x₂,...,x_s denote the distinct intersection numbers. The number s and the values of the i x ’s provide deep information about the structure of the design. For example, it is known that a 2-design with exactly one intersection number must be symmetric, and a 2-design with intersection numbers 0 and 1 only must be a nonsymmetric 2 − (υ, k,1) design (i.e., a finite projective plane or a Steiner system). These results are usually derived using linear algebra. A quasi-symmetric 2-design is a nontrivial 2-design with exactly two intersection numbers [5].

Recently, several studies on quasi-symmetric designs have been conducted, please refer to [3,4,6,7].

In this paper, we give a completely elementary proof of the fundamental equivalence concerning block intersection numbers. Our main result is the following theorem.

Theorem 1.1 (Intersection Number Property). For a 2 − (υ, k,λ ) design, the following are equivalent:
(i) The design is symmetric, i.e., b =υ.
(ii) Any two distinct blocks intersect in a constant number of points (necessarily λ).

The proof proceeds in two directions, each established as a lemma using only double-counting and elementary algebra. Lemma 3.1 shows that symmetric designs have constant intersection number, while Lemma 4.1 shows that if a design has constant intersection number, then it must be symmetric. Together they yield the desired characterization.

For any fixed block B₀, let the intersections with the other b−1 blocks be The total number of unordered pairs of points in B₀ is . Each such pair lies in 0 B and in exactly λ −1 other blocks, so

Consider the sum S of all block intersection sizes over unordered distinct block pairs:

Counting by points, each point belongs to r blocks and therefore contributes to S. Hence

Equations (2) and (3) will be the starting points for both directions of the proof.

Lemma 3.1. If a 2 − (υ, k,λ ) design is symmetric (b =υ ), then for any two distinct blocks B_i,B_j we have , the block intersection number is constant.

Proof. Assume the design is symmetric, so b =υ and r = k. From (1) we have λ (υ −1) = k (k −1). Fix a block B₀ and let the intersection sizes with the remaining n =υ −1 blocks be a₁,...,a_n. Equation (2) becomes

Counting incidences of points of B₀ with other blocks: each of the k points of B₀ lies in k − 1 other blocks, so

so a_i = λ for every i. Since B₀ was arbitrary, any two blocks in a symmetric design intersect in exactly λ points.

Lemma 4.1. If a 2 − (υ, k,λ ) design has constant block intersection numbers (i.e., ), for all i ≠ j then the design must be symmetric (b =υ ).

Proof. Assume there exists a constant μ such that for all distinct blocks B_i,B_j.

We shall show that b =υ.

From (3) we obtain

Under the constancy assumption, every block B₀ ≠ B satisfies x_B = μ. Equation (2) becomes

Substituting (7) into (8) and simplifying (cancelling a factor k) yields

Now eliminate λ using the basic relation Substitute this into the right-hand side of (9):

The left-hand side of (9) is Hence (9) is equivalent to

Using υ =bk/r we express everything in terms of b, k, r. After straightforward algebraic manipulation, one obtains

Now suppose, for contradiction, that the design is nonsymmetric, i.e., b >υ. Then b −υ > 0 and k − r < 0. For b >υ > k > 2, one checks that the right-hand side is strictly negative, implying that the left-hand side of (10) is negative. But equation (10) requires this expression to be zero. This contradiction shows that our assumption b >υ is impossible. Therefore, we must have b =υ, i.e., the design is symmetric.

Proof. If the design is symmetric, Lemma 3.1 gives constant intersection number. Conversely, if the design has constant block intersection numbers, Lemma 4.1 forces b =υ, hence the design is symmetric.

Using only double-counting and elementary algebra, we have proved that a 2-design has constant block intersection numbers precisely when it is symmetric. The proof is self-contained and avoids any advanced machinery such as incidence matrices, eigenvalue theory, or other advanced tools. This elementary approach makes the result accessible to students and researchers with a basic background in combinatorics, and it provides a clear illustration of the power of combinatorial counting arguments.

The authors declare no conflicts of interest.

This work is supported by the National Natural Science Foundation of China (No:11801092).

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