# On Homeomorphisms of Lebesgue Stieltjies measure with Lebesgue Measure II

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# On Homeomorphisms of Lebesgue Stieltjies measure with Lebesgue Measure

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# Borel Isomorphisms II

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# Borel Isomorphisms I

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# The tree property and the failure of SCH at \alpeh_{\omega^2} IV

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The tree property at \kappa^+ states that every tree with height \kappa^+ and levels of size at most \kappa has an unbounded branch. There is a tension between the tree property and the Singular Cardinal Hypothesis (SCH). Woodin asked if the failure of SCH at a singular cardinal \kappa implies the tree property at \kappa^+. Recently Neeman answered this question in the negative. Here we show that this result can be obtained at small cardinals. In particular we will show that given \omega many supercompact cardinals there is a generic extension in which the tree property holds at \aleph_{\omega^2+1} and SCH fails at \aleph_{\omega^2}.

# The tree property and the failure of SCH at \alpeh_{\omega^2} III

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The tree property at \kappa^+ states that every tree with height \kappa^+ and levels of size at most \kappa has an unbounded branch. There is a tension between the tree property and the Singular Cardinal Hypothesis (SCH). Woodin asked if the failure of SCH at a singular cardinal \kappa implies the tree property at \kappa^+. Recently Neeman answered this question in the negative. Here we show that this result can be obtained at small cardinals. In particular we will show that given \omega many supercompact cardinals there is a generic extension in which the tree property holds at \aleph_{\omega^2+1} and SCH fails at \aleph_{\omega^2}.

# The tree property and the failure of SCH at \alpeh_{\omega^2} II

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The tree property at \kappa^+ states that every tree with height \kappa^+ and levels of size at most \kappa has an unbounded branch. There is a tension between the tree property and the Singular Cardinal Hypothesis (SCH). Woodin asked if the failure of SCH at a singular cardinal \kappa implies the tree property at \kappa^+. Recently Neeman answered this question in the negative. Here we show that this result can be obtained at small cardinals. In particular we will show that given \omega many supercompact cardinals there is a generic extension in which the tree property holds at \aleph_{\omega^2+1} and SCH fails at \aleph_{\omega^2}.

# The tree property and the failure of SCH at \alpeh_{\omega^2}

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# The Tree Property at $\aleph_{\omega+1}$ IV

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We show that given $\omega$ many supercompact cardinals, there is a

generic extension in which there are no Aronszajn trees at

$\aleph_{\omega+1}$. This is an improvement of the large cardinal

assumptions. The previous hypothesis was a huge cardinal and $\omega$ many

supercompact cardinals above it, in Magidor-Shelah.