Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)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 consider a dynamic stochastic process for constructing a graph in which at each step we independently decide, with probability , whether the graph receives a new vertex or a new edge between existing vertices. The connections are then made according to a preferential attachment rule. We prove that the random graph produced by this so-called generalized linear preferential (GLP) model at time t contains a complete subgraph whose vertex set cardinality is given by , where , for any small asymptotically almost surely.1661137149Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP) [2013/24928-2]Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq)FAPEMIG (Programa Pesquisador Mineiro) [PPM 00600/16]Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior (CAPES)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)