Cluster Expansions: Necessary and Sufficient Convergence Conditions

We prove a new convergence condition for the activity expansion of correlation functions in equilibrium statistical mechanics with possibly negative pair potentials. For non-negative potentials, criterion is an if and only condition. The formulated sign-flipped Kirkwood-Salsburg operator known conditions such as Koteck${\'y}$-Preiss Fern${\'a}$ndez-Procacci are easily recovered. In addition, we deduce sufficient hard-core systems $\mathbb R^d$ Z^d$ well abstract polymer systems. latter improves on criterion.