Different capacities of a digraph
The \(n\)-th Sperner power \(G^ n\) of a digraph \(G\) with vertex set \(V\) has vertex set \(V^ n\) and contains edges \((x_ 1x_ 2 \cdots x_ n, y_ 1y_ 2 \cdots y_ n)\) if and only if \(G\) contains edge \((x_ i, y_ i)\) for at least one \(i\), \(1 \leq i \leq n\). If \(C\) is a class of digraphs closed under Sperner products, let \(F(G^ n)\) denote the cardinality of the largest subgraph from \(C\) contained as an induced subgraph of \(G^ n\); then \(\lim_{n \to \infty}n^{ - 1} \cdot \log F (G^ n)\) may be called the \(C\)-capacity of \(G\). The authors derive upper and lower bounds for various capacity functions and investigate the conditions under which they are tight.
scientific article; zbMATH DE number 644728
scientific article