Analysis of size of the largest dense subgraph of random hypergraph

Random networks are often described using Erdos-Renyi model of random graph $G(n,p)$. The concept of graph density is often used in random network analysis. In this article, the maximal size of $c$-dense subgraph almost surely included in random graph $G(n,\frac 12)$ was evaluated. It was shown, that if $c<\frac 12$, then $G(n,\frac 12)$ is almost surely $c$-dense; the upper and lower bounds for the size of maximal $\frac 12$-dense subgraph almost surely included in $G(n,\frac 12)$ were determined; in case when $c>\frac 12$, the upper bound for the maximal size of $c$-dense subgraph almost surely included in $G(n,\frac 12)$ was attained.