Combinatorial Game Distributions of Steiner Systems

The $P$-position sets of some combinatorial games have special combinatorial structures. For example, the $P$-position set of the hexad game, first investigated by Conway and Ryba, is the block set of the Steiner system $S(5, 6, 12)$ in the shuffle numbering, denoted by $D_{\text{sh}}$. However, few games were known to be related to Steiner systems in this way. For a given Steiner system, we construct a game whose $P$-position set is its block set. By using constructed games, we obtain the following two results. First, we characterize $D_{\text{sh}}$ among the 5040 isomorphic $S(5, 6, 12)$ with point set $\{0, 1, ..., 11\}$. For each $S(5, 6, 12)$, our construction produces a game whose $P$-position set is its block set. From $D_{\text{sh}}$, we obtain the hexad game, and this game is characterized as the unique game with the minimum number of positions among the obtained 5040 games. Second, we characterize projective Steiner triple systems by using game distributions. Here, the game distribution of a Steiner system $D$ is the frequency distribution of the numbers of positions in games obtained from Steiner systems isomorphic to $D$. We find that the game distribution of an $S(t, t + 1, v)$ can be decomposed into symmetric components and that a Steiner triple system is projective if and only if its game distribution has a unique symmetric component.

Model theory of Steiner triple systems

A Steiner triple system (STS) is a set [Formula: see text] together with a collection [Formula: see text] of subsets of [Formula: see text] of size 3 such that any two elements of [Formula: see text] belong to exactly one element of [Formula: see text]. It is well known that the class of finite STS has a Fraïssé limit [Formula: see text]. Here, we show that the theory [Formula: see text] of [Formula: see text] is the model completion of the theory of STSs. We also prove that [Formula: see text] is not small and it has quantifier elimination, [Formula: see text], [Formula: see text], elimination of hyperimaginaries and weak elimination of imaginaries.

Decomposing Graphs

This chapter considers problems of whether a graph can be decomposed into certain other kinds of graphs, primarily cycles. It begins with a background on nineteenth-century mathematician Thomas Penyngton Kirkman and the problem he invented known as Kirkman's Schoolgirl Problem, stated as: How many triples can be formed with x symbols in such a way that no pair of symbols occurs more than once in the triple? This is followed by a discussion of the Steiner triple system, the relationship between cyclic decomposition problems and a problem called Alspach's Conjecture, graceful graphs, and the Graceful Tree Conjecture. The chapter concludes with an analysis of the puzzle dubbed Instant Insanity and how graphs can be utilized to solve it.

Cyclic and rotational latin hybrid triple systems

It is well known that given a Steiner triple system (STS) one can define a binary operation * upon its base set by assigning

Methods of Constructing and Enumerating the Steiner Triple System with Order 31

This paper describes the basic concept of constructing Steiner triple system of order and gives the definition edge matrix of a complete graph. It proposes a method of constructing Steiner triple system of order. The entire procedure of constructing Steiner triple system of order .It discusses the enumeration problem of Steiner triple system.

A Method of Constructing Steiner Triple System of Order 2t+1

The basic concept of constructing the Steiner triple systems of order is described. The detinition of edges matrix of the complete graph is given. A method of constructing Steiner triple system of order is proposed. The errtire procedure of constructing Steiner triple systems of orderand is presented respectively, The enumeration problem of Steiner triple systems of order is discussed [1-1.