On the consistency strength of 'Accessible' Jonsson Cardinals and of the Weak Chang Conjecture

Using the core model K we determine better lower bounds for the consistency strength of some combinatorial principles: I. Assume that A is a Jonsson cardinal which is ‘accessible’ in the sense that at least one of (l)-(4) holds: (1) A is a successor cardinal; (2) A = oE and 6 <A ; (3) A is singular of uncountable cofinality; (4) A is a regular but not weakly hyper-Mahlo.Then Ot exists. II. For A = P+ a successor cardinal we consider the weak Chang Conjecture, WCC(A), which is a consequence of the Chang transfer property (A+, A) j (A, p). III. If A = p+ao,, then WCC(A) implies the existence of 0”. IV. We can determine the consistency strength of wCC(o,). We include a relatively simple definition of the core model which together with the results of Dodd and Jensen suffices for our proofs.