Abstract: The strongly n-ξ-Gorenstein projective objects in triangulated categories are introduced, strongly n-ξ-Gorenstein-projective objects are closed under finite direct sums is proved and a simple description of an object with ξ-Gorenstein projective dimension at most n is given.

Key words: ξ-Gorenstein-projective object, strong n-ξ-Gorenstein-projective object, proper class of triangles