An extremal problem related to biplanes

The existence problem for biplanes has proved to be intractable: only finitely many are known. However, it can be turned into an extremal problem, on which some progress can be made. A biplane is a set of subsets (blocks) of {I, ... , n} such that (a) any two blocks meet in two points; (b) any two points lie in two blocks. It is easy to see that, for some integer k, every block of a biplane contains k points, and every point lies in k blocks; the number of points and the number of blocks are both equal to (~) + 1. Thus, a biplane is just a symmetric 2-design (SBIBD) with ,\ = 2. Only finitely many biplanes are known at present. The known examples have k = 3, 4, 5, 6, 9, 11 and 13, having respectively 4, 7, 11, 16, 37, 56 and 79 points. (See, for example, Beth, Jungnickel and Lenz [1], or Brouwer's chapter [2] in the Handbook of Combinatorics.) With current methods, there seems to be no hope of deciding whether or not infinitely many biplanes exist. In view of the difficulty of this question and the scarcity of examples, we can turn it into an extremal problem: What is the smallest number m of subsets (blocks) of {I, ... ,n} such that ( a) any two blocks meet in at most two points; (b) any two points lie in at least two blocks? It is the opposing inequalities which give this problem its particular subtlety. Note that, if we have a configuration which satisfies the conditions, then removing a point leaves one which still satisfies them; so the extremal m is a monotonic increasing function of n. This problem arose from a question in genetics, and was communicated to me by Gregory Gutin. It turns out that the application can be done more efficiently in Australasian Journal of Combinatorics 20(1999), pp.97-100 an entirely different way, using search techniques based on coding theory, which will not be discussed here. The following result gives some bounds for m. Theorem 1 Let m be the least number of subsets of {I, ... ,n} satisfying conditions (a) and (b) above. (i) m 2: n, with equality if and only a biplane with n points exists. (ii) m ~ (2 + o(l))n. Proof (i) Count incidences between point-pairs and block-pairs. If i is the number of such incidences, then 2G) ::; i, by (a), and 2(r;) 2: i, by (b); so the inequality follows. If equality holds, then both bounds are tight, so we have equality in both (a) and (b); that is, we have a biplane. (ii) Let n = q2 + q + 1, q a prime power, and let D be a planar difference set in Zj(n). This is a subset of Zj(n) of size q + 1, having the property that any non-zero element of Z j (n) has a unique representation as the difference of two elements of D. Equivalently, the translates of D are the lines of a projective plane on the point set Zj(n). Now it is a standard result that -D (and hence any translate of -D) is an oval in this projective plane; that is, meets any line in at most two points. (To see this, suppose that \(D + x) n (-D + y)\ 2: 2. By translation, we may assume that x = O. Then there exist d1l d2l d~, d~ E D such that d1 = -d~ + Y and d2 = -d~ + y. Thus, d1 d2 = d~ d~, and the difference set property shows that d1 = d~ and d2 = d~. If there were a third intersection, say d3 = -d~ + y, we would have d1 = d~ and d3 = d~, a contradiction.) Moreover, -D is itself a difference set, since it is the image of D under an automorphism of Zj(n); so its translates form another projective plane. N ow take all translates of D and D. Any two points of Z j (n) lie in one translate of D and one of D. The above remarks show that any two of these sets meet in at most two points. Now the gap between consecutive primes Pn and Pn+l is known to be o(Pn) (indeed, O(p~) for some c < 1). So by choosing q to be the smallest prime (power) such that q2 + q + 1 2: n, we obtain the stated result. The next result shows that, if n is just a little smaller than the number of points in a biplane, then a biplane with some points removed is optimal. Theorem 2 Suppose that k ;::: 4 and (k;l) + 1 < n < (~) + 1. Then the number m of sets required satisfies