Weak compactness and no partial squares

Abstract. We present a characterization of weakly compact cardinals in terms of generalized stationarity. We apply this characterization to construct a model with no partial square sequences. One of the striking features of weak compactness in the large cardinal hierarchy is the wide variety of characterizations of the concept. These characterizations meet many disparate areas of set theory, including metamathematics, combinatorics, and elementary embeddings, among others. In this paper we expand on this theme by characterizing weak compactness in terms of generalized stationarity. The property of Mahloness illustrates a way to define or characterize a large cardinal property: specify a naturally defined set, and then assert that the large cardinal property holds iff the set is stationary. For example, a cardinal κ is Mahlo iff the set of strongly inaccessible cardinals below κ is stationary. The first goal of the paper is to characterize weakly compact cardinals along the same lines. Let κ> ω be a regular cardinal. Let S be the set of N in Pκ(H(κ satisfying: (1) N∩κ is strongly inaccessible, (2) N<N∩κ ⊆ N, and (3) the transitiv