Let $T$ be a complete first-order theory over a finite relational language which is axiomatized by universal and existential sentences. It is shown that $T$ is almost trivial in the sense that the universe of any model of $T$ can be written $F \overset{\cdot}{\cup} I_1 \overset{\cdot}{\cup} I_2 \overset{\cdot}{\cup} \cdots \overset{\cdot}{\cup} I_n$, where $F$ is finite and $I_1, I_2,\ldots,I_n$ are mutually indiscernible over $F$. Some results about complete theories with $\exists\forall$-axioms over a finite relational language are also mentioned.