Partitions of the 4-subsets of a 13-set into disjoint projective planes

✍ Scribed by Leo G. Chouinard II

• Publisher: Elsevier Science
• Year: 1983
• Tongue: English
• Weight: 422 KB
• Volume: 45
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(83)90045-6

✦ Synopsis

For I G t < k CI u. let S(t, k, u) denote a Steiner system and let Pr, (u) be the set of all k-subsets of theset {i,2,..., u}. We partition PJ 13) into 55 mutually disjoint S(2.4, 13)'s (projective planes). This is the first known example of a complete partition of Pk(u) into disjoint S(t, k, u)'s for k 34 and a a 2.

1. Intmductiun and bahgruund

A Steiner system S(t, k, u), 1 s t < k < v, is a subset of P&J) such that each t-subset of the u-set is in exactly one of the &-sets. Arithmetically, when an S(t, k, V) exists, there can be no more than M(z, k, u) = (L-J mutually disjoint S(t, fc, v)'s. Let D(t, k, V) be the maximal number of mutually disjoint S(t, k, 2))'s. Note &(a) partitions into S(t, k, V)'S when D(t, k, u) = M(t, k, u). There has been significant attention in the literature to the problem of determining D(t, k, u); much but not all of this has focused on S(2,3, D)'s. The following includes most of what is known at least for small values of K If t = 1 (so 21 = ck), then Baranyai [l] shows that D(l, k, ck) = M( I, k, ck). There are also infinitely many cases where D(2,3, u) = M(2,3, u) (see Teirlinck [lo]). Also D(2,3,7) -a: D(3,4,8) = 2 < 5 = M(2,3,7) = M(3,4,8) (Cayley [2]); D(2,3,9) = 7 = M(2,3,9) (Kirkman [4]); D(3,4,10) = 5 ~7 = M(3,4,10) and D(4,5,11) = D&6,12) = 2 <7 = _M(4,5, 11) = M(5,6,12) (Kramer, Mesner [5]); and D(2,3, 13) = 11 = M(2,3, 13) (Denniston [3]; Kramer, Mesner [6]). Other results can bi: found in [7-g].

In this paper we show that D(2,4,13) = 55 = M(2,4, i3), which is the first known case where D(t, k, u) = M( t, k, v) with t s 2 and k > 4. Here the S(2,4,13)'s are projective planes of order 3. This case was brought to the author's attention by Spyros Magliveras, who had shown that D(2,4,13) 3 33. Earl Kramer was particularly helpful m providing perspective on the problem.