We introduce a variant of the Erdős–Rényi random graph where the number of vertices is random and follows a Poisson law. A very simple Markov property of the model entails that the Lukasiewicz exploration is made of independent Poisson increments. Using a vanilla Poisson counting process, this enables us to give very short proofs of classical results such as the phase transition for the giant component or the connectedness for the standard Erdős–Rényi model.