Mitchell's theorem on the approachability ideal states that it is consistent relative to a greatly Mahlo cardinal that there is no stationary subset of ω 2 ∩ cof(ω 1 ) in the approachability ideal I[ω 2 ]. In this paper we give a new proof of Mitchell's theorem, deriving it from an abstract framework of side condition methods.

Assuming the existence of a Mahlo cardinal, we construct a model in which there exists an ω 2 -Aronszajn tree, the ω 1 -approachability property fails, and every stationary subset of ω 2 ∩ cof(ω) reflects. This solves an open problem of [1].

We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals κ, updating some classical constructions in the process. This includes models of CSR(κ ++ ) ∧ TP(κ ++ ) (both with and without AP(κ ++ )) and models of the conjunctions SR(κ ++ ) ∧ wTP(κ ++ ) ∧ AP(κ ++ ) and ¬AP(κ ++ ) ∧ SR(κ ++ ) (the latter was originally obtained in joint work by Krueger and the first author [8], and is here given using different methods). Analogs of these results with the failure of SH(κ ++ ) are given as well. Finally, we obtain all of our results with an arbitrarily large 2 κ , applying recent joint work by Honzik and the third author.

We show that the Abraham-Rubin-Shelah Open Coloring Axiom is consistent with a large continuum, in particular, consistent with 2 ℵ 0 = ℵ 3 . This answers one of the main open questions from [2]. As in [2], we need to construct names for so-called preassignments of colors in order to add the necessary homogeneous sets. However, these names are constructed over models satisfying the CH. In order to address this difficulty, we show how to construct such names with very strong symmetry conditions. This symmetry allows us to combine them in many different ways, using a new type of poset called a partition product, and thereby obtain a model of this axiom in which 2 ℵ 0 = ℵ 3 .

We prove from the existence of a Mahlo cardinal the consistency of the statement that 2 ω = ω 3 holds and every stationary subset of ω 2 ∩ cof(ω) reflects to an ordinal less than ω 2 with cofinality ω 1 .Let us say that stationary set reflection holds at ω 2 if for any stationary set S ⊆ ω 2 ∩ cof(ω) there is an ordinal α ∈ ω 2 ∩ cof(ω 1 ) such that S ∩ α is stationary in α (that is, S reflects to α). In a classic forcing construction, Harrington and Shelah [3] proved the equiconsistency of stationary set reflection at ω 2 with the existence of a Mahlo cardinal. Specifically, if stationary set reflection holds at ω 2 , then ω1 fails, and hence ω 2 is a Mahlo cardinal in L. Conversely, if κ is a Mahlo cardinal, then the generic extension obtained by Lévy collapsing κ to become ω 2 and then iterating to kill the stationarity of nonreflecting sets satisfies stationary set reflection at ω 2 . The Harrington-Shelah argument is notable because the majority of stationary set reflection principles are derived by extending large cardinal elementary embeddings, and thus use large cardinal principles much stronger than the existence of a Mahlo cardinal.The original Harrington-Shelah model satisfies the generalized continuum hypothesis, and in particular, that 2 ω = ω 1 . Suppose we would like to obtain a model of stationary set reflection at ω 2 together with 2 ω = ω 2 . A natural construction would be to iterate forcing with countable support of length a weakly compact cardinal κ, alternating between adding reals and collapsing ω 2 to have size ω 1 . Such an iteration P would be proper, κ-c.c., collapse κ to become ω 2 , and satisfy that 2 ω = ω 2 . The fact that stationary set reflection holds in any generic extension V [G] by P follows from the ability to extend an elementary embedding j with critical point κ after forcing with the proper forcing j(P)/G over V [G].Consider the problem of obtaining a model satisfying stationary set reflection at ω 2 together with 2 ω > ω 2 . Since in that case not all reals would be added by the iteration collapsing κ to become ω 2 , extending the elementary embedding becomes more difficult. Indeed, in the model referred to in the previous paragraph, a stronger stationary set reflection principle holds, namely WRP(ω 2 ), which asserts that any stationary subset of [ω 2 ] ω reflects to [β] ω for some uncountable β < ω 2 , and by a result of Todorcević, WRP(ω 2 ) implies 2 ω ≤ ω 2 (see [5, Lemma 2.9]).In this paper we demonstrate that the cardinality of the continuum provides a natural separation between ordinary stationary set reflection and higher order