2,149 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
Counting Configurations in Designs
AbstractGiven a t-(v,Ā k,Ā Ī») design, form all of the subsets of the set of blocks. Partition this collection of configurations according to isomorphism and consider the cardinalities of the resulting isomorphism classes. Generalizing previous results for regular graphs and Steiner triple systems, we give linear equations relating these cardinalities. For any fixed choice of t and k, the coefficients in these equations can be expressed as functions of v and Ī» and so depend only on the design's parameters, and not its structure. This provides a characterization of the elements of a generating set for m-line configurations of an arbitrary design
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