On hypergraphs having evenly distributed subhypergraphs
A family of \(k\)-uniform hypergraphs of order \(n\) has property \(U(r)\) for \(r\) fixed if every \(k\)-uniform hypergraph of order \(r\) occurs as an induced subgraph about \(n^ r/2^{{r\choose k}}\) times asymptotically as \(n\to \infty\). It was known that \(U(2k)\) implies \(U(r)\) for all fixed \(r\). Now the authors have shown that this is best possible by constructing for each \(s\), \(1\leq s\leq 2k-1\), families for which \(U(s)\) does not imply \(U(s+1)\). Thus, the remarkable phenomenon that occurs for graphs \((k=2)\) also manifests itself for hypergraphs. There are a couple of misprints in the definition. Interchange \(E\) and \(E'\) on line 7, and in equation (1), the binomial coefficient should be \(r\) above \(k\).
scientific article; zbMATH DE number 428306