Forcing a □(κ)-like principle to hold at a weakly compact cardinal

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Hellsten [17] proved that when κ is Πn1-indescribable, the n-club subsets of κ provide a filter base for the Πn1-indescribability ideal, and hence can also be used to give a characterization of Πn1-indescribable sets which resembles the definition of stationarity: a set S⊆κ is Πn1-indescribable if and only if S∩C≠∅ for every n-club C⊆κ. By replacing clubs with n-clubs in the definition of □(κ), one obtains a □(κ)-like principle □n(κ), a version of which was first considered by Brickhill and Welch [7]. The principle □n(κ) is consistent with the Πn1-indescribability of κ but inconsistent with the Πn+11-indescribability of κ. By generalizing the standard forcing to add a □(κ)-sequence, we show that if κ is κ+-weakly compact and GCH holds then there is a cofinality-preserving forcing extension in which κ remains κ+-weakly compact and □1(κ) holds. If κ is Π21-indescribable and GCH holds then there is a cofinality-preserving forcing extension in which κ is κ+-weakly compact, □1(κ) holds and every weakly compact subset of κ has a weakly compact proper initial segment. As an application, we prove that, relative to a Π21-indescribable cardinal, it is consistent that κ is κ+-weakly compact, every weakly compact subset of κ has a weakly compact proper initial segment, and there exist two weakly compact subsets S0 and S1 of κ such that there is no β<κ for which both S0∩β and S1∩β are weakly compact.