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Partition Relation Equiconsistent with $\exists \kappa(o(\kappa) = \kappa^+)$

Wednesday 26th August 2015 - 14:00 to 14:30
INI Seminar Room 2
Preamble: In this work we deal with partition relations with infinite exponents under $ZFC$, hence all results are limited to definable functions.

In [78], M. Spector has proven, basically, that $\exists \kappa(o(\kappa) = 1)$ is equiconsistent with $\aleph_1 \rightarrow (\omega)^{\omega}_{\aleph_0}$. In [87], we were able to show that the result generalizes to $n = 2$; namely, $\exists \kappa(o(\kappa) = 2)$ is equiconsistent with $\aleph_1 \rightarrow (\omega^2)^{\omega^2}_{\aleph_0}$. Surprisingly at first sight, this property cannot be generalized further (for $n> 2$), and later on we were able to prove that $\aleph_1 \rightarrow (\omega^3)^{\omega^3}_{\aleph_0}$ is equiconsistent with $\exists \kappa(o(o(\kappa)) = 2)$.

The above lead us to a finer notion of homogeneity:

Definition: Weak Homogeneity is the partition property $\kappa \xrightarrow{\text{\tiny WH}}(\lambda)^{\eta}_{\mu}$ where the only considered subsequences of $\lambda$ are those that are created by removing (or, complementarily, collecting) only finitely many segments of $\lambda$.

Using week homogeneity we were able to prove the following for any ordinal $\alpha$ [87]: $\exists \kappa(o(\kappa) = \alpha)$ is equiconsistent with $\aleph_1 \xrightarrow{\text{\tiny WH}}(\omega^\alpha)^{\omega^\alpha}_{\aleph_0}$.

Later on we were able to characterize the consistency strength of $\exists \kappa(o(\kappa)~=~\kappa)$, and recently we have arrived at the main result of this paper:

$\exists \kappa(o(\kappa) = \kappa^+)$ is equiconsistent with $\aleph_1 \xrightarrow{\text{\tiny WH}}(\aleph_1)^{\aleph_1}_{\aleph_0}$


[78] M. Spector: Natural Sentences of Mathematics which are independent of $V = L$, $V = L^\mu$ etc., 1978 (preprint).

[87] Y.M. Kimchi: Dissertation, 1987, The Hebrew University of Jerusalem, Israel
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons