Abstract

We show that if there is no inner model with a Woodin cardinal and the Steel core model K exists, then every Jonsson cardinal is Ramsey in K, and every $\delta$-Jonsson cardinal is $\delta$-Erdos in K. In the absence of the Steel core model K we prove the same conclusion for any model L$[\mathscr{E}]$ such that either V = L$[\mathscr{E}]$ is the minimal model for a Woodin cardinal, or there is no inner model with a Woodin cardinal and V is a generic extension of L$[\mathscr{E}]$. The proof includes one lemma of independent interest: If V = L$[\mathscr{A}]$, where A $\subset$ $\kappa$ and $\kappa$ is regular, then L$_\kappa$[A] is a Jonsson algebra. The proof of this result, Lemma 2.5, is very short and entirely elementary.