Stable intersection of Cantor sets in higher dimension and robust homoclinic tangency of the largest codimension

In this paper, we construct (a) a pair of two regular Cantor sets in higher dimension which exhibits $C^1$-stable intersection and (b) hyperbolic basic set $C^2$-robust homoclinic tangency the largest codimension for any dimensional manifold, using blenders. The former implies that an analog Moreira's theorem on real line does not hold dimension. latter solves question posed by Barrientos A.Raibekas.