Shelah's eventual categoricity conjecture in universal classes: part I

We prove: $\mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in cardinals of arbitrarily high cofinality, then $K$ is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on the Hanf number and get rid of the cofinality restrictions: $\mathbf{Theorem}$ Let $K$ be an AEC with amalgamation. Assume that $K$ is fully $\operatorname{LS} (K)$-tame and short and has primes over sets of the form $M \cup \{a\}$. Write $H_2 := \beth_{\left(2^{\beth_{\left(2^{\operatorname{LS} (K)}\right)^+}}\right)^+}$. If $K$ is categorical in a $\lambda > H_2$, then $K$ is categorical in all $\lambda' \ge H_2$.