The first omitting cardinal for Magidority

An infinite cardinal λ is Magidor if and only if (Formula presented.). It is known that if λ is Magidor then (Formula presented.) for some α < λ, and the first such α is denoted by α _m (λ). In this paper we try to understand some of the properties of α _m (λ). We prove that α _m (λ) can be the successor of a supercompact cardinal, when λ is a Magidor cardinal. From this result we obtain the consistency of α _m (λ) being a successor of a singular cardinal with uncountable cofinality.