553 research outputs found
Uniform hypergraphs containing no grids
A hypergraph is called an rĆr grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A1, ..., Ar, B1, ..., Br} such that Aiā©Aj=Biā©Bj=Ļ for 1ā¤i<jā¤r and {pipe}Aiā©Bj{pipe}=1 for 1ā¤i, jā¤r. Three sets C1, C2, C3 form a triangle if they pairwise intersect in three distinct singletons, {pipe}C1ā©C2{pipe}={pipe}C2ā©C3{pipe}={pipe}C3ā©C1{pipe}=1, C1ā©C2ā C1ā©C3. A hypergraph is linear, if {pipe}Eā©F{pipe}ā¤1 holds for every pair of edges Eā F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For rā„. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs. Ā© 2013 Elsevier Ltd
Completion and deficiency problems
Given a partial Steiner triple system (STS) of order , what is the order
of the smallest complete STS it can be embedded into? The study of this
question goes back more than 40 years. In this paper we answer it for
relatively sparse STSs, showing that given a partial STS of order with at
most triples, it can always be embedded into a complete
STS of order , which is asymptotically optimal. We also obtain
similar results for completions of Latin squares and other designs.
This suggests a new, natural class of questions, called deficiency problems.
Given a global spanning property and a graph , we define the
deficiency of the graph with respect to the property to be
the smallest positive integer such that the join has property
. To illustrate this concept we consider deficiency versions of
some well-studied properties, such as having a -decomposition,
Hamiltonicity, having a triangle-factor and having a perfect matching in
hypergraphs.
The main goal of this paper is to propose a systematic study of these
problems; thus several future research directions are also given
Configurations of lines and models of Lie algebras
The automorphism groups of the 27 lines on the smooth cubic surface or the 28
bitangents to the general quartic plane curve are well-known to be closely
related to the Weyl groups of and . We show how classical
subconfigurations of lines, such as double-sixes, triple systems or Steiner
sets, are easily constructed from certain models of the exceptional Lie
algebras. For and we are lead to
beautiful models graded over the octonions, which display these algebras as
plane projective geometries of subalgebras. We also interpret the group of the
bitangents as a group of transformations of the triangles in the Fano plane,
and show how this allows to realize the isomorphism in terms of harmonic cubes.Comment: 31 page
Set-Codes with Small Intersections and Small Discrepancies
We are concerned with the problem of designing large families of subsets over
a common labeled ground set that have small pairwise intersections and the
property that the maximum discrepancy of the label values within each of the
sets is less than or equal to one. Our results, based on transversal designs,
factorizations of packings and Latin rectangles, show that by jointly
constructing the sets and labeling scheme, one can achieve optimal family sizes
for many parameter choices. Probabilistic arguments akin to those used for
pseudorandom generators lead to significantly suboptimal results when compared
to the proposed combinatorial methods. The design problem considered is
motivated by applications in molecular data storage and theoretical computer
science
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