Calderbank's short proof [3] requires the theory of association schemes and their ... Let (Q, 0S) be a tactical configuration with parameters v, k, b, r, in which any ...
AN INEQUALITY FOR TACTICAL CONFIGURATIONS PETER M. NEUMANN AND CHERYL E. PRAEGER
This note is devoted to proving a theorem and a generalisation of it which were originally suggested by our group-theoretical studies but which in fact are purely combinatorial. After it was accepted for publication, we learned that the theorems are not new, although our proofs appear to be substantially simpler than those appearing elsewhere. We had checked the literature on combinatorial designs, whereas it was in the theory of hypergraphs that the results had already emerged, expressed in a very different language (see [6, 7, 8] and references quoted there). The proofs given in those papers use a considerable amount of sophisticated machinery, however. Even Calderbank's short proof [3] requires the theory of association schemes and their Bose-Mesner algebras. With the permission of the Editors we have decided to proceed with the publication of our proofs because they use only elementary counting techniques. A tactical configuration consists of a finite set Q of 'points', a finite set 3d of ' blocks' and an incidence relation between them, so that all blocks are incident with the same number k of points, and all points are incident with the same number r of blocks. (See, for example, Dembowski [4, p. 4].) If v := \Cl\ and b := \38\, then v, k, b, r are known as the parameters of the configuration. Counting incident point-block pairs, one sees that vr = bk. In what follows we shall consider blocks to be sets of points and use suggestive set-theoretic notation correspondingly. Nevertheless, the definition does not exclude the possibility that a tactical configuration may have ' repeated' blocks in the sense that two or more blocks may be incident with precisely the same set of k points. Indeed, from a configuration (Q, 3fi) with parameters v, k, b, r, a configuration may be obtained simply by repeating each block ft times (compare Fischer [5, p. 52]). This new configuration has parameters v, k, fib, fir, and we shall refer to it as the fith multiple of (€1,38). Going in the opposite direction, starting from (Q, 3d) we define % to be 3d with repetitions deleted; equivalently, it is the collection of subsets of Q that occur as the underlying point-sets of blocks. Echoing Fisher, we shall call (Q, 3SQ) the primary configuration underlying (Q, 3#). Note, however, that although all of its blocks are A;-sets, (Q., 3SQ) need not itself be a tactical configuration. In the language of hypergraphs, (Q,^o) *s a ^-uniform hypergraph (see [6,8]). The main focus of this note concerns tactical configurations (Q, 38) in which every pair of blocks meet (in the sense that there is a point incident with both of them). It is easy to see that the same is then true of the primary configurations (Q, 3S0). And it is not at all difficult to prove that v < k2. In the language of hypergraphs, such tactical configurations are known as intersecting fc-uniform regular hypergraphs, and according to Z. Fiiredi (see [7, p. 158]), the conjecture that they have at most k2—k +1 points was made by P. Erdos and by B. Bollobas, and was first proved by L. Lovasz Received 27 February 1995; revised 3 August 1995. 1991 Mathematics Subject Classification 05B05, 05B25, 20B05, 05C65. Bull. London Math. Soc. 28 (1996) 471-475
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PETER M. NEUMANN AND CHERYL E. PRAEGER
in 1975. A second proof, which also characterised the extreme configurations, was given in 1981 by Fiiredi [7, Corollary 4] using the theory of fractional matchings and fractional transversals in hypergraphs. The following proof is quite elementary. THEOREM 1. Let (Q, 0S) be a tactical configuration with parameters v, k, b, r, in which any two blocks meet. Then v ^ k2 — k + 1. Furthermore, v = k2 — k+\ if and only if (Q, 0S) is a multiple of a projective plane of order k—\.
Proof. Choose a base-point co0eQ and define Q 0 :=Q-{co 0 },
Clearly, |Q0| = v-\,
\®x\ = r and |# 2 | = b-r.
For coeQ0 define
so that Xu is the number of blocks of (Q, $8) that contain both co and co0. Then
Next define S := {(co, F 1 ; F2) \ I \ e&v F 2 e # 2 , coe T, n T2} and find | S \ in two different ways. On the one hand, since any two blocks of (Q, @) meet, Fl f] F 2 # 0 for any rie3$l, F2e$2. Therefore certainly
\S\^WW = r(b-r).
(2)
r
On the other hand, |5| = T,wen0L( -D and so £ Xa- £ A«.
r(b-r)^r
coefi 0
coen o
From Cauchy's Inequality we have V
i •;
cuefi0
and so ^ W6 n o ^> (v- Wd^LY
V /
