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Uniform hypergraphs containing no grids. (English) Zbl 1278.05161
Summary: A hypergraph is called an \(r \times r\) grid if it is isomorphic to a pattern of \(r\) horizontal and \(r\) vertical lines, i.e., a family of sets \(\{A_1, \dots, A_r, B_1, \dots, B_r\}\) such that \(A_i\cap A_j=B_i\cap B_j=\emptyset \) for \(1\leq i<j\leq r\) and \(|A_i\cap B_j|=1\) for \(1\leq i,j\leq r\). Three sets \(C_1,C_2,C_3\) form a triangle if they pairwise intersect in three distinct singletons, \(|C_1\cap C_2|=|C_2\cap C_3|=|C_3\cap C_1|=1\), \(C_1\cap C_2\neq C_1\cap C_3\). A hypergraph is linear, if \(|E\cap F|\leq 1\) holds for every pair of edges \(E\neq F\). 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 \(r\geq 4\) our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs.

MSC:
05C65 Hypergraphs
05C42 Density (toughness, etc.)
05D05 Extremal set theory
11B25 Arithmetic progressions
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