Set Systems with L‐Intersections and k‐Wise L‐Intersecting Families

In this paper, by employing linear algebra methods we obtain the following main results:

• (i) Let and be two disjoint subsets of such that Suppose that is a family of subsets of such that for every pair and for every i. Then Furthermore, we extend this theorem to k‐wise L‐intersecting and obtain the corresponding result on two cross L‐intersecting families. These results show that Snevily's conjectures proposed by Snevily (2003) are true under some restricted conditions. This result also gets an improvement of a theorem of Liu and Hwang (2013).
• (ii) Let p be a prime and let and be two subsets of such that or and Suppose that is a family of subsets of [n] such that (1) for every pair (2) for every i. Then This result improves the existing upper bound substantially.

     

Detachments of Amalgamated 3‐Uniform Hypergraphs Factorization Consequences

A is a hypergraph obtained from by splitting some or all of its vertices into more than one vertex. Amalgamating a hypergraph can be thought of as taking , partitioning its vertices, then for each element of the partition squashing the vertices to form a single vertex in the amalgamated hypergraph . In this paper, we use Nash‐Williams lemma on laminar families to prove a detachment theorem for amalgamated 3‐uniform hypergraphs, which yields a substantial generalization of previous amalgamation theorems by Hilton, Rodger, and Nash‐Williams. To demonstrate the power of our detachment theorem, we show that the complete 3‐uniform n‐partite multihypergraph can be expressed as the union of k edge‐disjoint factors, where for , is ‐regular, if and only if:

1. for all ,
2. for each i, , and
3. .

     

Simple Signed Steiner Triple Systems

Let X be a v‐set, be a set of 3‐subsets (triples) of X, and be a partition of with . The pair is called a simple signed Steiner triple system, denoted by ST, if the number of occurrences of every 2‐subset of X in triples is one more than the number of occurrences in triples . In this paper, we prove that exists if and only if , , and , where and for , . © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 332–343, 2012     

‐Relative Difference Sets and Their Representations

We show that every ‐relative difference set D in relative to can be represented by a polynomial , where is a permutation for each nonzero a. We call such an f a planar function on . The projective plane Π obtained from D in the way of M. J. Ganley and E. Spence (J Combin Theory Ser A, 19(2) (1975), 134–153) is coordinatized, and we obtain necessary and sufficient conditions of Π to be a presemifield plane. We also prove that a function f on with exactly two elements in its image set and is planar, if and only if, for any .     

Finiteness Results in Triangle‐Free Quasi‐Symmetric Designs

Triangle‐free quasi‐symmetric 2‐ designs with intersection numbers ; and are investigated. Possibility of triangle‐free quasi‐symmetric designs with or is ruled out. It is also shown that, for a fixed x and a fixed ratio , there are only finitely many triangle‐free quasi‐symmetric designs. © 2012 Wiley Periodicals, Inc. J Combin Designs 00: 1‐6, 2012     

Decomposition of Complete Graphs into Isomorphic Complete Bipartite Graphs

A decomposition of a complete graph into disjoint copies of a complete bipartite graph is called a ‐design of order n. The existence problem of ‐designs has been completely solved for the graphs for , for , K_{2, 3} and K_{3, 3}. In this paper, I prove that for all , if there exists a ‐design of order N, then there exists a ‐design of order n for all (mod ) and . Giving necessary direct constructions, I provide an almost complete solution for the existence problem for complete bipartite graphs with fewer than 18 edges, leaving five orders in total unsolved.     

Constructions of Strictly m‐Cyclic and Semi‐Cyclic

An is a triple , where X is a set of points, is a partition of X into m disjoint sets of size n and is a set of 4‐element transverses of , such that each 3‐element transverse of is contained in exactly one of them. If the full automorphism group of an admits an automorphism α consisting of n cycles of length m (resp. m cycles of length n), then this is called m‐cyclic (resp. semi‐cyclic). Further, if all block‐orbits of an m‐cyclic (resp. semi‐cyclic) are full, then it is called strictly cyclic. In this paper, we construct some infinite classes of strictly m‐cyclic and semi‐cyclic , and use them to give new infinite classes of perfect two‐dimensional optical orthogonal codes with maximum collision parameter and AM‐OPPTS/AM‐OPPW property.     

Flag‐Transitive 2‐ Symmetric Designs with Sporadic Socle

Let be a nontrivial 2‐ symmetric design admitting a flag‐transitive, point‐primitive automorphism group G of almost simple type with sporadic socle. We prove that there are up to isomorphism six designs, and must be one of the following: a 2‐(144, 66, 30) design with or , a 2‐(176, 50, 14) design with , a 2‐(176, 126, 90) design with or , or a 2‐(14,080, 12,636, 11,340) design with .     

On the Enumeration of ‐Optimal and Minimax‐Optimal k‐Circulant Supersaturated Designs

In recent years, several methods have been proposed for constructing ‐optimal and minimax‐optimal supersaturated designs (SSDs). However, until now the enumeration problem of such designs has not been yet considered. In this paper, ‐optimal and minimax‐optimal k‐circulant SSDs with 6, 10, 14, 18, 22, and 26 runs, factors and are enumerated in a computer search. We have also enumerated all ‐optimal and minimax‐optimal k‐circulant SSDs with (mod 4) and . The computer search utilizes the fact that theses designs are equivalent to certain 1‐rotational resolvable balanced incomplete block designs. Combinatorial properties of these resolvable designs are used to restrict the search space.     

Uniform Semi‐Latin Squares and Their Schur‐Optimality

Let n and k be integers, with and . An semi‐Latin square S is an array, whose entries are k‐subsets of an ‐set, the set of symbols of S, such that each symbol of S is in exactly one entry in each row and exactly one entry in each column of S. Semi‐Latin squares form an interesting class of combinatorial objects which are useful in the design of comparative experiments. We say that an semi‐Latin square S is uniform if there is a constant μ such that any two entries of S, not in the same row or column, intersect in exactly μ symbols (in which case ). We prove that a uniform semi‐Latin square is Schur‐optimal in the class of semi‐Latin squares, and so is optimal (for use as an experimental design) with respect to a very wide range of statistical optimality criteria. We give a simple construction to make an semi‐Latin square S from a transitive permutation group G of degree n and order , and show how certain properties of S can be determined from permutation group properties of G. If G is 2‐transitive then S is uniform, and this provides us with Schur‐optimal semi‐Latin squares for many values of n and k for which optimal semi‐Latin squares were previously unknown for any optimality criterion. The existence of a uniform semi‐Latin square for all integers is shown to be equivalent to the existence of mutually orthogonal Latin squares (MOLS) of order n. Although there are not even two MOLS of order 6, we construct uniform, and hence Schur‐optimal, semi‐Latin squares for all integers . & 2012 Wiley Periodicals, Inc. J. Combin. Designs 00: 1–13, 2012     

Turyn‐Type Sequences: Classification,Enumeration, and Construction

Turyn‐type sequences, , are quadruples of ‐sequences , with lengths , respectively, where the sum of the nonperiodic autocorrelation functions of and twice that of is a δ‐function (i.e., vanishes everywhere except at 0). Turyn‐type sequences are known to exist for all even n not larger than 36. We introduce a definition of equivalence to construct a canonical form for in general. By using this canonical form, we enumerate the equivalence classes of for . We also construct the first example of Turyn‐type sequences .     

The Asymptotic Existence of Resolvable Group Divisible Designs

A group divisible design (GDD) is a triple which satisfies the following properties: (1) is a partition of X into subsets called groups; (2) is a collection of subsets of X, called blocks, such that a group and a block contain at most one element in common; and (3) every pair of elements from distinct groups occurs in a constant number λ blocks. This parameter λ is usually called the index. A k‐GDD of type is a GDD with block size k, index , and u groups of size g. A GDD is resolvable if the blocks can be partitioned into classes such that each point occurs in precisely one block of each class. We denote such a design as an RGDD. For fixed integers and , we show that the necessary conditions for the existence of a k‐RGDD of type are sufficient for all . As a corollary of this result and the existence of large resolvable graph decompositions, we establish the asymptotic existence of resolvable graph GDDs, G‐RGDDs, whenever the necessary conditions for the existence of ‐RGDs are met. We also show that, with a few easy modifications, the techniques extend to general index. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 112–126, 2013     

Cycle Frames of Complete Multipartite Multigraphs ‐ III

For two graphs G and H their wreath product has vertex set in which two vertices and are adjacent whenever or and . Clearly, , where is an independent set on n vertices, is isomorphic to the complete m‐partite graph in which each partite set has exactly n vertices. A 2‐regular subgraph of the complete multipartite graph containing vertices of all but one partite set is called partial 2‐factor. For an integer λ, denotes a graph G with uniform edge multiplicity λ. Let J be a set of integers. If can be partitioned into edge‐disjoint partial 2‐factors consisting cycles of lengths from J, then we say that has a ‐cycle frame. In this paper, we show that for and , there exists a ‐cycle frame of if and only if and . In fact our results completely solve the existence of a ‐cycle frame of .     

Semiframes with Block Size Three and Odd Group Size

A ‐semiframe of type is a ‐GDD of type , , in which the collection of blocks can be written as a disjoint union where is partitioned into parallel classes of and is partitioned into holey parallel classes, each holey parallel class being a partition of for some . A ‐SF is a ‐semiframe of type in which there are p parallel classes in and d holey parallel classes with respect to . In this paper, we shall show that there exists a (3, 1)‐SF for any if and only if , , , and .     

Large Cross‐Free Sets in Steiner Triple Systems

A cross‐free set of size m in a Steiner triple system is three pairwise disjoint m‐element subsets such that no intersects all the three ‐s. We conjecture that for every admissible n there is an STS(n) with a cross‐free set of size which if true, is best possible. We prove this conjecture for the case , constructing an STS containing a cross‐free set of size 6k. We note that some of the 3‐bichromatic STSs, constructed by Colbourn, Dinitz, and Rosa, have cross‐free sets of size close to 6k (but cannot have size exactly 6k). The constructed STS shows that equality is possible for in the following result: in every 3‐coloring of the blocks of any Steiner triple system STS(n) there is a monochromatic connected component of size at least (we conjecture that equality holds for every admissible n). The analog problem can be asked for r‐colorings as well, if and is a prime power, we show that the answer is the same as in case of complete graphs: in every r‐coloring of the blocks of any STS(n), there is a monochromatic connected component with at least points, and this is sharp for infinitely many n.     

The Existence and Construction of (K5∖e)‐Designs of Orders 27, 135, 162, and 216

The problem of the existence of a decomposition of the complete graph into disjoint copies of has been solved for all admissible orders n, except for 27, 36, 54, 64, 72, 81, 90, 135, 144, 162, 216, and 234. In this paper, I eliminate 4 of these 12 unresolved orders. Let Γ be a ‐design. I show that divides 2^k3 for some and that . I construct ‐designs by prescribing as an automorphism group, and show that up to isomorphism there are exactly 24 ‐designs with as an automorphism group. Moreover, I show that the full automorphism group of each of these designs is indeed . Finally, the existence of ‐designs of orders 135, 162, and 216 follows immediately by the recursive constructions given by G. Ge and A. C. H. Ling, SIAM J Discrete Math 21(4) (2007), 851–864.     

Three‐Phase Barker Arrays

A 3‐phase Barker array is a matrix of third roots of unity for which all out‐of‐phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two‐dimensional 3‐phase Barker arrays have size 2 × 2 or 3 × 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3‐phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a double‐exponentially growing arithmetic function T such that no 3‐phase Barker array of size with exists for all . For example, , , and . When both dimensions are divisible by 3, the existence problem is settled completely: if a 3‐phase Barker array of size exists, then .     

The Mendelsohn Triple Systems of Order 13

A triple cyclically contains the ordered pairs , , , and no others. A Mendelsohn triple system of order v, or , is a set V together with a collection of ordered triples of distinct elements from V, such that and each ordered pair with is cyclically contained in exactly λ ordered triples. By means of a computer search, we classify all Mendelsohn triple systems of order 13 with ; there are 6 855 400 653 equivalence classes of such systems.     

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

A pseudo‐hyperoval of a projective space , q even, is a set of subspaces of dimension such that any three span the whole space. We prove that a pseudo‐hyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles that admit a point‐primitive, line‐transitive automorphism group with a point‐regular abelian normal subgroup. Specifically, we show that is flag‐transitive and isomorphic to , where is either the regular hyperoval of PG(2, 4) or the Lunelli–Sce hyperoval of PG(2, 16).     

Partitions of V(n,q) into 2‐ and s‐Dimensional Subspaces

Let denote a vector space of dimension n over the field with q elements. A set of subspaces of V is a (vector space) partition of V if every nonzero element of V is contained in exactly one subspace in . Suppose that is a partition of V with subspaces of dimension for . Then we call the type of the partition . Which possible types correspond to actual partitions is in general an open question. We prove that for any odd integer and for any integer , the existence of partitions of across a suitable range of types guarantees the existence of partitions of of essentially all the types for any integer . We then apply this result to construct new classes of partitions of V. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 467‐482, 2012