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The Ostaszewski square and homogeneous Souslin trees

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Abstract

Assume GCH and let λ denote an uncountable cardinal. We prove that if □λ holds, then this may be witnessed by a coherent sequence 〈C α|α < λ+〉 with the following remarkable guessing property

For every sequence 〈A i | i < λ〉 of unbounded subsets of λ +, and every limit θ < λ, there exists some α < λ + such that otp(C α)=θ and the (i + 1) th -element of C α is a member of A i , for all i < θ.

As an application, we construct a homogeneous λ +-Souslin tree from □λ + CHλ, for every singular cardinal λ.

In addition, as a by-product, a theorem of Farah and Veličković, and a theorem of Abraham, Shelah and Solovay are generalized to cover the case of successors of regulars.

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Correspondence to Assaf Rinot.

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Rinot, A. The Ostaszewski square and homogeneous Souslin trees. Isr. J. Math. 199, 975–1012 (2014). https://doi.org/10.1007/s11856-013-0065-0

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  • DOI: https://doi.org/10.1007/s11856-013-0065-0

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