  We prove the existence of a large complete subgraph w.h.p. in a preferential
attachment random graph process with an edge-step. That is, we prove that the
random graph $G_{t}$ produced by the so-called GLP model at time $t$ contains a
complete subgraph of order $t^\alpha$, where $\alpha =
(1-\varepsilon)\frac{1-p}{2-p}$, $\varepsilon$ is any number such that
$0<\varepsilon<1$, and $0<p<1$ is a parameter of the model.
