Tame failures of the unique branch hypothesis and models of ADℝ + Θ is regular

Abstract

In this paper, we show that the failure of the unique branch hypothesis ([Formula: see text]) for tame iteration trees implies that in some homogenous generic extension of [Formula: see text] there is a transitive model [Formula: see text] containing [Formula: see text] such that [Formula: see text] is regular. The results of this paper significantly extend earlier works from [Non-tame mice from tame failures of the unique branch bypothesis, Canadian J. Math. 66(4) (2014) 903–923; Core models with more Woodin cardinals, J. Symbolic Logic 67(3) (2002) 1197–1226] for tame trees.