Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Special Tree Number   Define the special tree number, denoted $\mathfrak{st}$, to be the least size
of a tree of height $\omega_1$ which is neither special nor has a cofinal
branch. This cardinal had previously been studied in the context of fragments
of $\mathsf{MA}$ but in this paper we look at its relation to other, more
typical, cardinal characteristics. Classical facts imply that $\aleph_1 \leq
\mathfrak{st} \leq 2^{\aleph_0}$, under Martin's Axiom $\mathfrak{st} =
2^{\aleph_0}$ and that $\mathfrak{st} = \aleph_1$ is consistent with
$\mathsf{MA}({\rm Knaster}) + 2^{\aleph_0} = \kappa$ for any regular $\kappa$
thus the value of $\mathfrak{st}$ is not decided by $\mathsf{ZFC}$ and in fact
can be strictly below essentially all well studied cardinal characteristics. We
show that conversely it is consistent that $\mathfrak{st} = 2^{\aleph_0} =
\kappa$ for any $\kappa$ of uncountable cofinality while ${\rm non}(\mathcal M)
= \mathfrak{a} = \mathfrak{s} = \mathfrak{g} = \aleph_1$. In particular
$\mathfrak{st}$ is independent of the lefthand side of Cicho\'{n}'s diagram,
amongst other things. The proof involves an in depth study of the standard ccc
forcing notion to specialize (wide) Aronszajn trees, which may be of
independent interest.

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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an
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The categories of this paper are: cs
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The categories of this paper are: acc-phys
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The categories of this paper are: eess
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: supr-con
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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc