Triple System Of Order Research Articles

Steiner triple systems of order 19 constructed from the Steiner triple system of order 9

SynopsisA standard method of constructing Steiner triple systems of order 19 from the Steiner triple system of order 9 gives rise to 212 different such systems. It is shown that there are just three isomorphism classes amongst these systems. Representatives of each isomorphism class are described and the orders of their automorphism groups are determined.

Coloring Steiner Triple Systems

In this paper, several results on the chromatic number of Steiner triple systems are established. A Steiner triple system is a simple 3-uniform hypergraph in which every pair of vertices is connected by exactly one 3-edge. Among other things, we prove that for any $k\geqq 3$ there exists an $n_k $ such that for all admissible $v \geqq n_k $ there exists a k-chromatic Steiner triple systems of order v. In addition we prove that for all $v \geqq 49$ there exists a 4-chromatic Steiner triple system of order v. An estimate of $n_k $ is also established, namely, $c_1 k^2 \log k > n_k > c_2 k^2 $.

A Class of Steiner Triple Systems of Order 21 and Associated Kirkman Systems

A Class of Steiner Triple Systems of Order 21 and Associated Kirkman Systems

The structure of certain triple systems

For each prime power q ≡ 7 ( mod 12 ) q \equiv 7 \pmod {12} , there is a triple system of order q whose automorphism group is transitive on unordered pairs. The object of this paper is to study these systems. This is done by analyzing how pairs of elements are linked. The linkage of a and b consists of a triple (a, b, c) and of some cycles in which adjacent pairs of elements form triples alternately with a and with b. Because of the transitivity, the lengths of the cycles will be independent of the choice of a and b. Using a computer, the linkage between two elements was determined for each q > 1000 q > 1000 . Some curious facts concerning the lengths of the cycles were uncovered; for example, the number of cycles of length greater than 4 is even. The systems of prime order p > 1000 p > 1000 were found to have no proper subsystems of order greater than 3. In the remaining case, q = 343 q = 343 , there are subsystems of orders 7 and 49, and all subsystems of the same order are isomorphic. For no q with 7 > q > 1000 7 > q > 1000 is the automorphism group doubly transitive. Finally, some general results are proved. The cycles of lengths 4 and 6 are determined. Using this result, it is shown that there can be no subsystem of order 7 or 9, except for the subsystems of order 7 when q is a power of 7. Hence, by a theorem of Marshall Hall, the automorphism group cannot be doubly transitive, except possibly when q is a power of 7. (Added August 1974. In a postscript, it is shown that the automorphism group is not doubly transitive in this case either.)

The Structure of Certain Triple Systems

For each prime power $q \equiv 7 \pmod {12}$, there is a triple system of order q whose automorphism group is transitive on unordered pairs. The object of this paper is to study these systems. This is done by analyzing how pairs of elements are linked. The linkage of a and b consists of a triple (a, b, c) and of some cycles in which adjacent pairs of elements form triples alternately with a and with b. Because of the transitivity, the lengths of the cycles will be independent of the choice of a and b. Using a computer, the linkage between two elements was determined for each $q < 1000$ . Some curious facts concerning the lengths of the cycles were uncovered; for example, the number of cycles of length greater than 4 is even. The systems of prime order $p < 1000$ were found to have no proper subsystems of order greater than 3. In the remaining case, $q = 343$, there are subsystems of orders 7 and 49, and all subsystems of the same order are isomorphic. For no q with $7 < q < 1000$ is the automorphism group doubly transitive. Finally, some general results are proved. The cycles of lengths 4 and 6 are determined. Using this result, it is shown that there can be no subsystem of order 7 or 9, except for the subsystems of order 7 when q is a power of 7. Hence, by a theorem of Marshall Hall, the automorphism group cannot be doubly transitive, except possibly when q is a power of 7. (Added August 1974. In a postscript, it is shown that the automorphism group is not doubly transitive in this case either.)

On the number of steiner triple systems

We obtain a new lower estimate for the number N(n) of nonisomorphic Steiner triple systems of order n: $$N(n) \geqslant n^{\frac{{n^2 }}{{12}} - O\left( {\frac{{n^2 }}{{logn}}} \right)} .$$ This makes it possible to show that log N(n) is of order n2 log n.

Construction of Steiner Triple Systems Having Exactly One Triple in Common

A Steiner triple system is a pair (Q, t) where Q is a set and t a collection of three element subsets of Q such that each pair of elements of Q belong to exactly one triple of t. The number |Q| is called the order of the Steiner triple system (Q, t). It is well-known that there is a Steiner triple system of order n if and only if n ≡ 1 or 3 (mod 6). Therefore in saying that a certain property concerning Steiner triple systems is true for all n it is understood that n ≡ 1 or 3 (mod 6). Two Steiner triple systems (Q, t1) and (Q, t2) are said to be disjoint provided that t1 ∩ t2 = Ø. Recently, Jean Doyen has shown the existence of a pair of disjoint Steiner triple systems of order n for every n ≧ 7 [1].

Determination of Steiner Triple Systems of Order 15

Determination of Steiner Triple Systems of Order 15