----------------------------------------------------------------------------------------------- Math 6023 Topics in Discrete Math: Design and Graph Theory Fall 2007 ------------------------------------------------------------------------------------------------ Design Theory Notes 1: The purpose of these notes is to supplement the text by providing comments, examples and alternate proofs of some statements. They are meant to be used in conjunction with the text and are not a substitute for it. All numerical references are to statements and formulae in, P.J. Cameron and J.H. van Lint, Designs, Graphs, Codes and their Links, Cambridge University Press, 1991. (1.1) DEFINITION. A t-design with parameters (v,k,) (or a t-(v,k,) design) is a pair D = (X, B), where X is a set of "points" of cardinality v, and B is a collection of k-element subsets of X called "blocks", with the property that any t points of X are contained in precisely blocks. Example: A 3-(8,4,1) design is given by X = {1,2, ... ,8} and the collection B of 4-subsets: {1,2,5,6} {3,4,7,8} {1,3,5,7} {2,4,6,8} {1,4,5,8} {2,3,6,7} {1,2,3,4} {5,6,7,8} {1,2,7,8} {3,4,5,6} {1,3,6,8} {2,4,5,7} {1,4,6,7} {2,3,5,8} since any 3-set of points of X is contained in exactly 1 block. You may also verify that this same design is a 2-(8,4,3) design, since every pair of elements of X appear in exactly 3 blocks. Comment: The non-degeneracy condition given in the text ( v ≥ k ≥ t ) is considerably weaker than what is generally presented in the literature ( v > k > t > 0). The cases of equality are non- interesting designs. If v = k then every block contains all of the elements of X, and with the further restriction of no repeated blocks, there would be only one block. If k = t then every t-subset of X would have to be a block, this type of design, in which all the k-sets are blocks, is called a full combinatorial design and it is generally excluded from consideration. For example, in Proposition (1.2), the condition "not every k-set of points is incident with a block" excludes only the full combinatorial designs. Notation: It is nearly universal that the number of blocks in a t-design is denoted by b and the number of blocks containing a particular point, which is the same for any point and called the replication number, is denoted by r. However, it is useful to have a notation for the number of blocks containing a particular set of i points (with i ≤ t) since, as we shall see, this number does not depend upon the set chosen. Thus, we define i to be the number of blocks containing an i- set, for 0 ≤ i ≤ t. It then follows that 0 = b, 1 = r, and t = . The constancy of these parameters is proved in the next proposition. (1.4) Proposition. Let (S) be the number of blocks containing a given set S of s points in a t- (v,k,) design, where 0 ≤ s ≤ t. Then k−s v−s S = . t−s t−s Pf. Consider the set of ordered pairs (B, Z) where B is a block of the design containing S and Z is a set of t-s points of B not in S. We will count the number of such ordered pairs in two ways. First, the number of first coordinates is (S) and for each of these we can obtain an appropriate Z by choosing t-s points from the k-s points of B that are not in S. We thus obtain the LHS of the formula. On the other hand, we can first choose a set Z by selecting t-s points from the points of the design which are not in S (there are v-s of these). We now count the number of blocks that contain both S and Z. Since | S ∪ Z | = t, there are exactly blocks containing the union, giving the count on the RHS. Notice that in the above proof only the size of the set S was used, so (S) does not depend upon S, only on its size, s. Thus, (S) = s and is a constant for each s, 0 ≤ s ≤ t. Here is a computational alternative to the proof of: (1.12) Lemma. If I and J are the identity and all-1 matrices of order n, then det( xI + yJ) = (x + yn)xn-1. Pf. We are trying to determine the determinant xy y y y ⋯ y y xy y y ⋯ y y y xy y ⋯ y detx I y J = det y y y xy ⋯ y ∣ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ∣ y y y y ⋯ xy We can evaluate this by subtracting the first column from each of the other columns and then adding each row to the first row to obtain the following: xny 0 0 0 ⋯ 0 y x 0 0 ⋯ 0 y 0 x 0 ⋯ 0 detx I y J = det . y 0 0 x ⋯ 0 ∣ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ∣ y 0 0 0 ⋯ x The result now follows. Notation: 2-designs were the basis for the definition of t-designs and have a long history. These are the most studied designs and have gone under a number of different names; block designs, BIBD's( Balanced Incomplete Block Design), or just designs. More is known about 2-designs than any other type of t-design, so notice the specialization that occurs starting with theorem (1.14). (1.14) Theorem. (Fisher's Inequality) In a 2-design with k < v, we have b ≥ v. (1.17) DEFINITION. A 2-design is called square if b = v (and thus r = k). Comment: The term square is highly non-standard. The authors are trying to get the community of design theorists to give up some poorly chosen terminology and replace it by reasonable choices. This has been tried in the past with no avail. The term that they are trying to displace here is "symmetric", but this is well entrenched in the literature even though there is very little symmetry in a symmetric design. Examples: A square 2-(7,3,1) design is given by X = {1,2, ... ,7} and B consisting of the blocks: {1,2,4} {2,3,5} {3,4,6} {4,5,7} {5,6,1} {6,7,2} {7,1,3}. A square 2-(4,3,2) design is given by X = {1,2,3,4} and B consisting of the blocks: {1,2,3} {2,3,4} {3,4,1} {4,1,2}. (1.18) DEFINITION. A duality of a square design D is an isomorphism from D to its dual. It can be described as a pair of bijections : X B and : B X such that x ∈ B if and only if B∈ x , for all x ∈ X and B ∈ B. Note the corrections in the above definition. Example: In the square 2-(4,3,2) design above, we can label the blocks as follows: A = {1,2,3} B = {2,3,4} C = {3,4,1} and D = {4,1,2}. Now we can define a duality by: : X B : B X 1 B A 4 2 C B 1 3 D C 2 4 A D 3 It is easily verified that the condition is valid, and this pair of bijections defines a duality. What's more, since = -1 this duality is a polarity. Comment: The non-existence of the 2-(111,11,1) design, also known as the projective plane of order 10, was proved in 1989 by the most computer intensive combinatorial search that has been done to date. It required over a year of CPU time on a Cray (time and machine donated by NSA) to finish the search. Special Classes of Square Designs Definition: A projective plane of order n is a symmetric 2-(n2+n+1, n+1,1) design. They are known to exist whenever n is a prime power. A special type of projective plane is obtained as a restriction to 2 dimensions of the following construction. Definition: Projective Geometries PG(n,q). Let V be an (n+1)-dimensional vector space over the finite field GF(q). PG(n,q) is the set of all non-trivial vector subspaces of V. An i-flat is a subspace of vector space dimension i+1. 0-flats, 1-flats, 2-flats and (n-1)-flats are called points, lines, planes and hyperplanes respectively. The points and i-flats of PG(n,q) for 1 ≤ i ≤ n-1 form 2-designs. The points and lines design is a 2-((qn+1 – 1)/(q-1), q+1,1) design (a Steiner system), and the points and hyperplanes form a 2- ((qn+1 – 1)/(q-1), (qn – 1)/(q-1), (qn-1 – 1)/(q-1)) symmetric design. When n = 2, this last design is a projective plane. (1.23) Theorem. For a finite projective plane D, the following conditions are equivalent: (a) D is the point-line design of PG(2,q) for some prime power q; (b) D satisfies Desargues' Theorem; (c) D satisfies Pappus' Theorem; (d) Aut(D) is 2-transitive on the points of D. Comment: The finiteness assumption is necessary, and is missing from the statement in the text. In infinite projective planes, the Desargues and Pappus theorems are not equivalent, while Pappus implies Desargues the converse is not true (but it is true for finite planes, a consequence of Wedderburn's Theorem on finite division rings).