On the consistency strength of ''accessible'' Jonsson cardinals and of the weak Chang conjecture (Q761448)

It is shown that if \(\lambda\) is a Jónsson cardinal, and either \(\lambda =\omega_{\xi}\) with \(\xi <\lambda\), or \(\lambda\) is regular, not weakly hyper-Mahlo, or \(\lambda\) is singular of uncountable cofinality, then \(O^+\) holds. wCC(\(\lambda)\) denotes the following property weaker than Chang's conjecture: every function \(\lambda\) \(\to \lambda\) has a Galvin-Hajnal rank less than \(\lambda^+\). If \(\lambda =\rho^+\geq \omega_ 2\) then wCC(\(\lambda)\) implies the existence of \(0^+\), and the exact consistency strength of \(wCC(\omega_ 1)\) is given (the existence of a weakening of \(\omega_ 1\)-Erdős cardinals). The proof uses elementary properties of the core model as well as the extended covering lemma for K, and other upshots of the Dodd-Jensen theory.