Holey Steiner pentagon systems

In this article, it is shown that the necessary conditions for the existence of a holey Steiner pentagon system (HSPS) of type hⁿ are also sufficient, except possibly for the following cases: (1) when n = 15, and h ≡ 1 or 5 (mod 6) where h ≢ 0 (mod 5), or h = 9; and (2) (h, n) ∈ {(6, 6), (6, 36), (15, 19), (15, 23), (15, 27), (30, 18), (30, 22), (30, 24)}. Moreover, the results of this article guarantee the analogous existence results for group divisible designs (GDDs) of type hⁿ with block-size k = 5 and index λ = 2. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 41–56, 1999     

Super‐simple resolvable balanced incomplete block designs with block size 4 and index 3

The necessary conditions for the existence of a super‐simple resolvable balanced incomplete block design on v points with k = 4 and λ = 3, are that v ≥ 8 and v ≡ 0 mod 4. These conditions are shown to be sufficient except for v = 12. © 2003 Wiley Periodicals, Inc.     

Super‐simple resolvable balanced incomplete block designs with block size 4 and index 2

The necessary conditions for the existence of a super‐simple resolvable balanced incomplete block design on v points with block size k = 4 and index λ = 2, are that v ≥ 16 and . These conditions are shown to be sufficient. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 341–356, 2007     

Some rigid Steiner 5‐designs

Hitherto, all known non‐trivial Steiner systems S(5, k, v) have, as a group of automorphisms, either PSL(2, v−1) or PGL(2, (v−2)/2) × C₂. In this article, systems S(5, 6, 72), S(5, 6, 84) and S(5, 6, 108) are constructed that have only the trivial automorphism group. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:392–400, 2010     

Constructions for cyclic 3‐designs and improved results on cyclic Steiner quadruple systems

This paper gives some recursive constructions for cyclic 3‐designs. Using these constructions we improve Grannell and Griggs's construction for cyclic Steiner quadruple systems, and many known recursive constructions for cyclic Steiner quadruple systems are unified. Finally, some new infinite families of cyclic Steiner quadruple systems are obtained. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:178‐201, 2011     

The number of the non‐full‐rank Steiner triple systems

The $p$‐rank of a Steiner triple system (STS) $B$ is the dimension of the linear span of the set of characteristic vectors of blocks of $B$, over GF $\left(p\right)$. We derive a formula for the number of different STSs of order $v$ and given $2$‐rank $r_2$, $r_2<v$, and a formula for the number of STSs of order $v$ and given $3$‐rank $r_3$, $r_3<v?1$. Also, we prove that there are no STSs of $2$‐rank smaller than $v$ and, at the same time, $3$‐rank smaller than $v?1$. Our results extend previous study on enumerating STSs according to the rank of their codes, mainly by Tonchev, V.A. Zinoviev, and D.V. Zinoviev for the binary case and by Jungnickel and Tonchev for the ternary case.     

Constructions for resolvable and related designs

Methods are given for constructing block designs, using resolvable designs. These constructions yield methods for generating resolvable and affine designs and also affine designs with affine duals. The latter are transversal designs or semi-regular group divisible designs with ₁=0 whose duals are also designs of the same type and parameters. The paper is a survey of some old and some recent constructions.     

Existence of 3‐chromatic Steiner quadruple systems

A Steiner quadruple system of order v (briefly SQS (v)) is a pair (X, ), where X is a v‐element set and is a set of 4‐element subsets of X (called blocks or quadruples), such that each 3‐element subset of X is contained in a unique block of . The chromatic number of an SQS(v)(X, ) is the smallest m for which there is a map such that for all , where . The system (X, ) is equitably m‐chromatic if there is a proper coloring with minimal m for which the numbers differ from each other by at most 1. Linek and Mendelsohn showed that an equitably 3‐chromatic SQS(v) exists for v ≡ 4, 8, 10 (mod 12), v ≥ 16. In this article we show that an equitably 3‐chromatic SQS(v) exists for v ≡ 2 (mod 12) with v > 2. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 469–477, 2007     

Uniform two‐class regular partial Steiner triple systems

A 2‐class regular partial Steiner triple system is a partial Steiner triple system whose points can be partitioned into 2‐classes such that no triple is contained in either class and any two points belonging to the same class are contained in the same number of triples. It is uniform if the two classes have the same size. We provide necessary and sufficient conditions for the existence of uniform 2‐class regular partial Steiner triple systems.     

Large monochromatic components in 3‐edge‐colored Steiner triple systems

It is known that in any r‐coloring of the edges of a complete r‐uniform hypergraph, there exists a spanning monochromatic component. Given a Steiner triple system on n vertices, what is the largest monochromatic component one can guarantee in an arbitrary 3‐coloring of the edges? Gyárfás proved that $\left(2n+3\right)/3$ is an absolute lower bound and that this lower bound is best possible for infinitely many $n$. On the other hand, we prove that for almost all Steiner triple systems the lower bound is actually $\left(1?o\left(1\right)\right)n$. We obtain this result as a consequence of a more general theorem which shows that the lower bound depends on the size of a largest 3‐partite hole (ie, disjoint sets $X_1,X_2,X_3$ with $|X_1|=|X_2|=|X_3|$ such that no edge intersects all of $X_1,X_2,X_3$) in the Steiner triple system (Gyárfás previously observed that the upper bound depends on this parameter). Furthermore, we show that this lower bound is tight unless the structure of the Steiner triple system and the coloring of its edges are restricted in a certain way. We also suggest a variety of other Ramsey problems in the setting of Steiner triple systems.     

Infinite classes of anti‐mitre and 5‐sparse Steiner triple systems

In this paper, we present three constructions for anti‐mitre Steiner triple systems and a construction for 5‐sparse ones. The first construction for anti‐mitre STSs settles two of the four unsettled admissible residue classes modulo 18 and the second construction covers such a class modulo 36. The third construction generates many infinite classes of anti‐mitre STSs in the remaining possible orders. As a consequence of these three constructions we can construct anti‐mitre systems for at least 13/14 of the admissible orders. For 5‐sparse STS(υ), we give a construction for υ ≡ 1, 19 (mod 54) and υ ≠ 109. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 237–250, 2006     

4‐*GDDs(3n) and generalized Steiner systems GS(2, 4, v, 3)

Generalized Steiner systems GS(2, k, v, g) were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size g + 1 with minimum Hamming distance 2k ? 3, in which each codeword has length v and weight k. As to the existence of a GS(2, k, v, g), a lot of work has been done for k = 3, while not so much is known for k = 4. The notion k‐*GDD was first introduced and used to construct GS(2, 3, v, 6). In this paper, singular indirect product (SIP) construction for GDDs is modified to construct GS(2, 4, v, g) via 4‐*GDDs. Furthermore, it is proved that the necessary conditions for the existence of a 4‐*GDD(3ⁿ), namely, n ≡ 0, 1 (mod 4) and n ≥ 8 are also sufficient. The known results on the existence of a GS(2, 4, v, 3) are then extended. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 381–393, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10047     

Constructions for anti‐mitre Steiner triple systems

In this paper, we present a recursive construction for anti‐mitre Steiner triple systems. Furthermore, we present another construction of anti‐mitre STSs by utilizing 5‐sparse ones. © 2004 Wiley Periodicals, Inc.     

1‐Rotational Steiner triple systems over arbitrary groups

Phelps and Rosa introduced the concept of 1‐rotational Steiner triple system, that is an STS(ν) admitting an automorphism consisting of a fixed point and a single cycle of length ν ? 1 [Discrete Math. 33 ( 12 ), 57–66]. They proved that such an STS(ν) exists if and only if ν ≡ 3 or 9 (mod 24). Here, we speak of a 1‐rotational STS(ν) in a more general sense. An STS(ν) is 1‐rotational over a group G when it admits G as an automorphism group, fixing one point and acting regularly on the other points. Thus the STS(ν)'s by Phelps and Rosa are 1‐rotational over the cyclic group. We denote by ??_1r, ??_1r, ??_1r, ??_1r, the spectrum of values of ν for which there exists a 1‐rotational STS(ν) over an abelian, a cyclic, a dicyclic, and an arbitrary group, respectively. In this paper, we determine ??_1r and find partial answers about ??_1r and ??_1r. The smallest 1‐rotational STSs have orders 9, 19, 25 and are unique up to isomorphism. In particular, the only 1‐rotational STS(25) is over SL₂(3), the special linear group of dimension 2 over Z₃. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 215–226, 2001     

Quasi‐embeddings of Steiner triple systems,or Steiner triple systems of different orders with maximum intersection

In this paper, we present a conjecture that is a common generalization of the Doyen–Wilson Theorem and Lindner and Rosa's intersection theorem for Steiner triple systems. Given u, v ≡ 1,3 (mod 6), u < v < 2u + 1, we ask for the minimum r such that there exists a Steiner triple system such that some partial system can be completed to an STS , where |?| = r. In other words, in order to “quasi‐embed” an STS(u) into an STS(v), we must remove r blocks from the small system, and this r is the least such with this property. One can also view the quantity (u(u ? 1)/6) ? r as the maximum intersection of an STS(u) and an STS(v) with u < v. We conjecture that the necessary minimum r = (v ? u) (2u + 1 ? v)/6 can be achieved, except when u = 6t + 1 and v = 6t + 3, in which case it is r = 3t for t ≠ 2, or r = 7 when t = 2. Using small examples and recursion, we solve the cases v ? u = 2 and 4, asymptotically solve the cases v ? u = 6, 8, and 10, and further show for given v ? u > 2 that an asymptotic solution exists if solutions exist for a run of consecutive values of u (whose required length is no more than v ? u). Some results are obtained for v close to 2u + 1 as well. The cases where ≈ 3u/2 seem to be the hardest. © 2004 Wiley Periodicals, Inc.     

Existence of non‐resolvable Steiner triple systems

We consider two well‐known constructions for Steiner triple systems. The first construction is recursive and uses an STS(v) to produce a non‐resolvable STS(2v + 1), for v ≡ 1 (mod 6). The other construction is the Wilson construction that we specify to give a non‐resolvable STS(v), for v ≡ 3 (mod 6), v > 9. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 16–24, 2005.     

From Squashed 6‐Cycles to Steiner Triple Systems

Squashed 6‐cycle systems are introduced as a natural counterpart to 2‐perfect 6‐cycle systems. The spectrum for the latter has been determined previously in [5]. We determine completely the spectrum for squashed 6‐cycle systems, and also for squashed 6‐cycle packings.     

Decomposable super‐simple NRBIBDs with block size 4 and index 6

Necessary conditions for the existence of a super‐simple, decomposable, near‐resolvable ‐balanced incomplete block design (BIBD) whose 2‐component subdesigns are both near‐resolvable ‐BIBDs are (mod ) and . In this paper, we show that these necessary conditions are sufficient. Using these designs, we also establish that the necessary conditions for the existence of a super‐simple near‐resolvable ‐RBIBD, namely (mod ) and , are sufficient. A few new pairwise balanced designs are also given.     

Group divisible designs with block size four and type gum1—III

We deal with group divisible designs (GDDs) that have block size four and group type $g^u{m}^1$, where $g\equiv 2$ or 4 (mod 6). We show that the necessary conditions for the existence of a 4‐GDD of type $g^u{m}^1$ are sufficient when $g$ = 14, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 58, 62, 68, 76, 88, 92, 100, 104, 116, 124, 136, 152, 160, 176, 184, 200, 208, 224, 232, 248, 272, 304, 320, 368, 400, 448, 464 and 496. Using these results we go on to show that the necessary conditions are sufficient for $g=2^t{q}^s$, $q$ = 19, 23, 25, 29, 31, $s$, $t=1,2,\dots$, as well as for $g=2^{t}q$, $q$ = 2, 5, 7, 11, 13, 17, $t=1,2,\dots$, with possible exceptions $56^9{m}^1$, $80^9{m}^1$ and $112^9{m}^1$ for a few large values of $m$.     

Group divisible designs with block size 4 and type

Building upon the work of Wei and Ge (Designs, Codes, and Cryptography 74, 2015), we extend the range of positive integer parameters g, u, and m for which group divisible designs with block size 4 and type are known to exist. In particular, we show that the necessary conditions for the existence of these designs when and are sufficient in the following cases: , with one exception, 2⁶5¹, , and .