The design of the century

Abstract

ABSTRACT. We construct a 2-chromatic Steiner system S(2, 4, 100) in which every block contains three points of one colour and one point of the other colour.The existence of such a design has been open for over 25 years. Steiner system S(t, k, v) is an ordered pair (V, B) where V is a set of cardinality v, the base set, and B is a collection of k-subsets of V , the blocks, which collectively have the property that every t-element subset of V is contained in precisely one block. Elements of V are called points. In this paper we are principally concerned with the case in which t = 2 and k = 4. Steiner systems S(2, 4, v) exist if and only if v ≡ 1 or 4 (mod 12), [4]; such values of v are called admissible. Given a Steiner system S(2, 4, v), we may ask whether it is possible to colour each point of the base set V with one of two colours, say red or blue, so that no block is monochromatic. A Steiner system S(2, 4, v) having this property is said to be 2-chromatic or to have a blocking set. It was shown in [5] that 2-chromatic S(2, 4, v)s exist for all admissible v with the possible exception of three values, v = 37, 40 and 73. Existence for these three values was established in [3]. Perhaps we should also remark here that it is known that for all v ≥ 25 there exists a Steiner system S(2, 4, v) which is not 2-chromatic, [8].In a 2-chromatic S(2, 4, v) let c and v − c be the cardinalities of the red and blue colour classes, respectively. If b 1 , b 2 and b 3 are the numbers of blocks with 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 05B05. K e y w o r d s: Steiner system, colouring, 2-chromatic, blocking set.

New type B colorableS(2, 4,v) designs

“…The closely related problem of embedding Steiner triple systems into S (2, 4, v) systems is discussed at length in [6]. A type B 2-colorable S(2, 4, 100) was recently constructed by the authors and appears in [3]. In the current article we give constructions for other systems listed in Table I; specifically for v = 61 and 109.…”

“…In the current article we give constructions for other systems listed in Table I; specifically for v = 61 and 109. Also we briefly describe the system for v = 100 from [3] for completeness. We make no claim that any of the systems are unique up to isomorphism for their types.…”