Steiner Triple Systems and Existentially Closed Graphs (original) (raw)

Steiner Triple Systems Intersecting in Pairwise Disjoint Blocks

The Electronic Journal of Combinatorics, 2004

Two Steiner triple systems (X,calA)(X,{\cal A})(X,calA) and (X,calB)(X,{\cal B})(X,calB) are said to intersect in mmm pairwise disjoint blocks if ∣calAcapcalB∣=m|{\cal A}\cap{\cal B}|=m∣calAcapcalB∣=m and all blocks in calAcapcalB{\cal A}\cap{\cal B}calAcapcalB are pairwise disjoint. For each vvv, we completely determine the possible values of mmm such that there exist two Steiner triple systems of order vvv intersecting in mmm pairwise disjoint blocks.

The Fine Intersection Problem for Steiner Triple Systems

Graphs and Combinatorics, 2008

The intersection of two Steiner triple systems (X, A) and (X, B) is the set A ∩ B. The fine intersection problem for Steiner triple systems is to determine for each v, the set I(v), consisting of all possible pairs (m, n) such that there exist two Steiner triple systems of order v whose intersection I satisfies | ∪A∈I A| = m and |I| = n. We show that for v ≡ 1 or 3 (mod 6), |I(v)| = Θ(v 3), where previous results only imply that |I(v)| = Ω(v 2).

Embedding Partial Steiner Triple Systems

Proceedings of the London Mathematical Society, 1980

We prove that a partial Steiner triple system 8 of order n can be embedded in a Steiner triple system T of any given admissible order greater than 4w. Furthermore, if G(S), the missing-edge graph of S, has the property that A(G)<ri(n + l)l and \E(G)\ then # can be embedded in a Steiner triple system of order 2n +1, provided that 2w +1 is admissible.

Independent sets in Steiner triple systems

Ars Combinatoria, 2004

A set of points in a Steiner triple system (STS(v)) is said to be independent if no three of these points occur in the same block. In this paper we derive for each k 8 a closed formula for the number of independent sets of cardinality k in an STS(v). We use the formula to prove that every STS(21) has

Silver block intersection graphs of Steiner systems

Arxiv preprint arXiv: …, 2010

For a block design D, a series of block intersection graphs G i , or i-BIG(D), i = 0,. .. , k is defined in which the vertices are the blocks of D, with two vertices adjacent if and only if the corresponding blocks intersect in exactly i elements. A silver graph G is defined with respect to a maximum independent set of G, called an α-set. Let G be an r-regular graph and c be a proper (r + 1)-coloring of G. A vertex x in G is said to be rainbow with respect to c if every color appears in the closed neighborhood N [x] = N (x) ∪ {x}. Given an α-set I of G, a coloring c is said to be silver with respect to I if every x ∈ I is rainbow with respect to c. We say G is silver if it admits a silver coloring with respect to some I. Finding silver graphs is of interest, for a motivation and progress in silver graphs see [7] and [15]. We investigate conditions for 0-BIG(D) and 1-BIG(D) of Steiner 2-designs D = S(2, k, v) to be silver.

A Construction of Almost Steiner Systems

Journal of Combinatorial Designs, 2013

Let n, k, and t be integers satisfying n > k > t ≥ 2. A Steiner system with parameters t, k, and n is a k-uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for t ≥ 6. In this note we prove that for every k > t ≥ 2 and sufficiently large n, there exists an almost Steiner system with parameters t, k, and n; that is, there exists a k-uniform hypergraph on n vertices such that every set of t distinct vertices is covered by either one or two edges.

Graph-theoretic perspective on a special class of Steiner Systems

Eprint Arxiv 1410 5855, 2014

We study S(t−1,t,2t)S(t-1,t,2t)S(t−1,t,2t), which is a special class of Steiner systems. Explicit constructions for designing such systems are developed under a graph-theoretic platform where Steiner systems are represented in the form of uniform hypergraphs. The constructions devised are then used to study the 222-coloring properties of these uniform hypergraphs.

Strong Ramsey theorems for Steiner systems

Transactions of the American Mathematical Society, 1987

It is shown that the class of partial Steiner (fc, Z)-systems has the edge Ramsey property, i.e., we prove that for every partial Steiner (k, i)-system Q there exists a partial Steiner (fc, Z)-system)i such that for every partition of the edges of H into two classes one can find an induced monochromatic copy of Q. As an application we get that the class of all graphs without cycles of lengths 3 and 4 has the edge Ramsey property. This solves a longstanding problem in the area.

On determining when small embeddings of partial Steiner triple systems exist

Journal of Combinatorial Designs, 2020

A partial Steiner triple system of order u is a pair (U, A) where U is a set of u elements and A is a set of triples of elements of U such that any two elements of U occur together in at most one triple. If each pair of elements occur together in exactly one triple it is a Steiner triple system. An embedding of a partial Steiner triple system (U, A) is a (complete) Steiner triple system (V, B) such that U ⊆ V and A ⊆ B. For a given partial Steiner triple system of order u it is known that an embedding of order v 2u + 1 exists whenever v satisfies the obvious necessary conditions. Determining whether "small" embeddings of order v < 2u + 1 exist is a more difficult task. Here we extend a result of Colbourn on the NP-completeness of these problems. We also exhibit a family of counterexamples to a conjecture concerning when small embeddings exist.