Constraints on $N_c$ in Extensions of the Standard Model

We consider a class of theories involving an extension of the Standard Model gauge group to an {\it a priori} arbitrary number of colors, $N_c$, and derive constraints on $N_c$. One motivation for this is the string theory landscape. For two natural classes of embeddings of this $N_c$-extended Standard Model in a supersymmetric grand unified theory, we show that requiring unbroken electromagnetic gauge invariance, asymptotic freedom of color, and three generations of quarks and leptons forces one to choose $N_c=3$. Similarly, we show that for a theory combining the $N_c$-extended Standard Model with a one-family SU(2)$_{TC}$ technicolor theory, only the value $N_c=3$ is allowed.