Properties of Steiner triple systems of order 21

Properties of the 62,336,617 Steiner triple systems of order 21 with a non-trivial automorphism group are examined. In particular, there are 28 which have no parallel class, six that are 4-chromatic, five that are 3-balanced, 20 that avoid the mitre, 21 that avoid the crown, one that avoids the hexagon and two that avoid the prism. All systems contain the grid. None have a block intersection graph that is 3-existentially closed.Comment: 12 page

Ramsey goodness of <i>k</i>-uniform paths, or the lack thereof

Given a pair of k-uniform hypergraphs (G,H), the Ramsey number of (G,H), denoted by R(G,H), is the smallest integer n such that in every red/blue-colouring of the edges of Kn(k) there exists a red copy of G or a blue copy of ~H. Burr showed that, for any pair of graphs (G,H), where G is large and connected, R(G,H)≥(v(G)−1)(χ(H)−1)+σ(H), where σ(H) stands for the minimum size of a colour class over all proper χ(H)-colourings of H. We say that G is H-good if R(G,H) is equal to the general lower bound. Burr showed that, for any graph ~H, every sufficiently long path is H-good.Our goal is to explore the notion of Ramsey goodness in the setting of k-uniform hypergraphs. We demonstrate that, in stark contrast to the graph case, k-uniform ℓ-paths are not H-good for a large class of k-graphs. On the other hand, we prove that long loose paths are always at least asymptotically H-good for every H and derive lower and upper bounds that are best possible in a certain sense.In the 3-uniform setting, we complement our negative result with a positive one, in which we determine the Ramsey number asymptotically for pairs containing a long tight path and a 3-graph H when H belongs to a certain family of hypergraphs. This extends a result of Balogh, Clemen, Skokan, and Wagner for the Fano plane asymptotically to a much larger family of 3-graphs

Uniquely 3-colourable Steiner triple systems

A Steiner triple system (STS(v)) is said to be 3-balanced if every 3-colouring of it is equitable; that is, if the cardinalities of the colour classes differ by at most one. A 3-colouring, φ, of an STS(v) is unique if there is no other way of 3-colouring the STS(v) except possibly by permuting the colours of φ. We show that for every admissible v⩾25, there exists a 3-balanced STS(v) with a unique 3-colouring and an STS(v) which has a unique, non-equitable 3-colouring

Configurations and colouring problems in block designs

A Steiner triple system of order v (STS(v)) is called x-chromatic if x is the smallest number of colours needed to avoid monochromatic blocks. Amongst our results on colour class structures we show that every STS (19) is 3- or 4-chromatic, that every 3-chromatic STS(19) has an equitable 3-colouring (meaning that the colours are as uniformly distributed as possible), and that for all admissible v > 25 there exists a 3-chromatic STS(v) which does not admit an equitable 3-colouring. We obtain a formula for the number of independent sets in an STS(v) and use it to show that an STS(21) must contain eight independent points. This leads to a simple proof that every STS(21) is 3- or 4-chromatic. Substantially extending existing tabulations, we provide an enumeration of STS trades of up to 12 blocks, and as an application we show that any pair of STS(15)s must be 3-1-isomorphic. We prove a general theorem that enables us to obtain formulae for the frequencies of occurrence of configurations in triple systems. Some of these are used in our proof that for v > 25 no STS(u) has a 3-existentially closed block intersection graph. Of specific interest in connection with a conjecture of Erdos are 6-sparse and perfect Steiner triple systems, characterized by the avoidance of specific configurations. We describe two direct constructions that produce 6-sparse STS(v)s and we give a recursive construction that preserves 6-sparseness. Also we settle an old question concerning the occurrence of perfect block transitive Steiner triple systems. Finally, we consider Steiner 5(2,4, v) designs that are built from collections of Steiner triple systems. We solve a longstanding problem by constructing such systems with v = 61 (Zoe’s design) and v = 100 (the design of the century)