Is there any "singlet state" for 3 or more spin 1/2 particles?
Every system with $N$ or more electrons lies in a Hilbert space $H=H_{\text{space}} \otimes H_{\text{spin}}$, with $H_{\text{space}}=H_{\text{space}}^{1}\otimes\cdots\otimes H_{\text{space}}^{N}$ and $H_{\text{spin}}=H_{\text{spin}}^{1}\otimes\cdots\otimes H_{\text{spin}}^{N}$, $H^i$ being the $i$-th particle space. So the system has a state $|\Psi\rangle = |\Psi\rangle_{\text{space}} \otimes |\Psi\rangle_{\text{spin}} \in H$.
What I couldn't come up with is an antisymmetric spin ket $|\Psi\rangle_{\text{spin}}$ when there was more than 3 electrons. This would mean that the only way of antisymmetrizing $|\Psi\rangle$, for $N\geq 3$, is by antisymmetrizing only the spatial part. I think it's strange, since for $N=2$ we do have an antisymmetric spin ket (the singlet state), so why wouldn't be such kets for $N\geq 3$? 656