Abstract: We say that a hyperbolic set $\Lambda$ exhibits a $C^1$-robust homoclinic tangency if, for this set and all its close $C^1$ continuations, there is an orbit of non-transverse intersection in $W^u(\Lambda)\cap W^s(\Lambda)$. Let $f$ be a $C^r$ $ (r=1, \dots, \infty, \omega)$ diffeomorphism of a manifold with dimension >2, and let $f$ have a homoclinic tangency to a hyperbolic periodic point $O$. We prove that, if the central dynamics near $O$ are at least three dimensional and are not sectionally dissipative, then $f$ is accumulated in the $C^r$ topology by diffeomorphisms having hyperbolic sets with uncountably many $C^1$-robust homoclinic tangencies.
