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

Theorem 0.2Let K be an AEC with amalgamation. Assume that K is fully LS(K)-tame and short and has primes over sets of the form M∪{a}. Write H2:=ℶ(2ℶ(2LS(K))+)+. If K is categorical in a λ>H2, then K is categorical in all λ′≥H2.