Kunen tree and Martin tree - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T05:30:13Z http://mathoverflow.net/feeds/question/90010 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90010/kunen-tree-and-martin-tree Kunen tree and Martin tree alephomega 2012-03-02T03:44:22Z 2013-01-23T02:58:56Z <p>Do we know under which conditions the Kunen tree (Recall the Kunen tree provides an analysis of the equivalence classes of functions $f: \omega_1 \to \omega_1$ with respect to the normal measure $W^1_1$ on $\omega_1$ under $AD$) is homogeneous, if it is? Do we know if and under which hypothesis the Martin tree (it is a generalization of the Kunen tree to all the projective hierarchy) is homogeneous? Any reference treating their homogeneity will be appreciated. Thx.</p> http://mathoverflow.net/questions/90010/kunen-tree-and-martin-tree/119614#119614 Answer by alephomega for Kunen tree and Martin tree alephomega 2013-01-23T02:58:56Z 2013-01-23T02:58:56Z <p>Let me answer my question which I stumbled on while looking at my profile. The answer is yes. Actually assuming $AD$, if $\kappa$ is less than the supremum of the Suslin cardinals then every tree $T$ on $\omega \times \kappa$ is weakly homogeneous (Martin and Woodin).</p> <p>Also assuming that we have a Woodin $\delta$, a tree $T$ on $\omega \times \alpha$, $\alpha$ some ordinal, then there is a $\kappa &lt; \delta$ such that in the generic extension $V[G]$ where $G$ is generic for $Col(\omega, \kappa)$, $T$ is $&lt; \delta$ weakly homogeneous. </p> <p>One can modify the assumption and just assume that there is a pair $T$ and $U$ of $\delta^+$ absolutely complementing trees.</p>