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Morris and Saxton used the method of containers to bound the number of ‐vertex graphs with edges containing no ‐cycles, and hence graphs of girth more than . We consider a generalization to ‐uniform hypergraphs. The girth of a hypergraph is the minimum such that there exist distinct vertices and hyperedges with for all ...
Let be a graph of density p on n vertices. Following Erdős, Łuczak, and Spencer, an m‐vertex subgraph H of G is called full if H has minimum degree at least . Let denote the order of a largest full subgraph of G. If is a nonnegative integer, define
Erdős, Łuczak, and Spencer proved that for ,
In this article, we prove the following lower bound: for ,
Furthermore,...
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