We investigate numerically the three-dimensional three-state Potts models with nearest-neighbor (NN) ferromagnetic coupling and next-nearest-neighbor (NNN) antiferromagnetic coupling of relative strength γ on ${\mathit{L}}^3$ lattices with L=12, 16, 20, 32, and 40. For all the γ values that we studied, 0.0≤γ≤0.8, we find indications of a first-order phase transition between the ordered and disordered phases. In the neighborhood of γ=0.25, the latent heat becomes rather small, making it necessary to use still larger lattices to rule out a higher-order phase transition at γ=0.25. We also studied the boundary regimes of the three different ordered phases and find no criticality along them, thus suggesting a lack of criticality in these extended Potts models.

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