  We prove existence of twisted K\"ahler-Einstein metrics in big cohomology
classes, using a divisorial stability condition. In particular, when $-K_X$ is
big, we obtain a uniform Yau-Tian-Donaldson existence theorem for
K\"ahler-Einstein metrics. To achieve this, we build up from scratch the theory
of Fujita-Odaka type delta invariants in the transcendental big setting, using
pluripotential theory. We do not use the K-energy in our arguments, and our
techniques provide a simple roadmap to prove Yau-Tian-Donaldson existence
theorems for K\"ahler-Einstein type metrics, that only needs convexity of the
appropriate Ding energy. As an application, we give a simplified proof of
Li-Tian-Wang's existence theorem in the log Fano setting.
