We demonstrate that the elastic stress relaxation mechanism in micrometre-sized, highly mismatched heterostructures may be enhanced by employing patterned substrates in the form of necked pillars, resulting in a significant reduction of the dislocation density. Compositionally graded Si1-xGex crystals were grown by low energy plasma enhanced chemical vapour deposition, resulting in tens of micrometres tall, three-dimensional heterostructures. The patterned Si(001) substrates consist of micrometre-sized Si pillars either with the vertical {110} or isotropically under-etched sidewalls resulting in narrow necks. The structural properties of these heterostructures were investigated by defect etching and transmission electron microscopy. We show that the dislocation density, and hence the competition between elastic and plastic stress relaxation, is highly influenced by the shape of the substrate necks and their proximity to the mismatched epitaxial material. The SiGe dislocation density increases monotonically with the crystal width but is significantly reduced by the substrate under-etching. The drop in dislocation density is interpreted as a direct effect of the enhanced compliance of the under-etched Si pillars, as confirmed by the three-dimensional finite element method simulations of the elastic energy distribution. Published by AIP Publishing.