For the Grassmann manifold \(\widetilde G_{n,4}\) of oriented 4-planes in \(\mathbb{R}^{n}\) nofull description of its cohomology ring with coefficients in the two element field \(\mathbb {Z}_{2}\) is available. It is known however that it contains a subring that can be identifiedwith a quotient of a polynomial ring by a certain ideal. Examining this quotientring by means of Gröbner bases we are able to determine the \(\mathbb {Z}_{2}\) -cup-length of \(\widetilde G_{n,4}\) for \(n=2^t,2^t-1,2^t-2\) for all \(t \geq 4\) .
