Suppose the first-order language with equality and a unary function symbol $S$.
Want to find a sentence $\gamma $ such that no finite structure for this language satisfies $\gamma $ and some infinite structure $B$ for this language satisfies $\gamma.$ (finite means $|B|$ was a finite set.)
I solved it by making sure $S$ was injective and there was an element in $B$ could never be mapped. Thus any finite set could not satisfy the relation since its range would be smaller than its domain, i.e. $\forall x \forall y \wedge \rightarrow =SxSy =xy \neg \exists z =0Sz $
However, I couldn't figure out if there is any "weaker sentence" or "weakest sentence" $\gamma$ such that $\gamma$ satisfies the question, and whenever $\vDash_B \forall x \forall y \wedge \rightarrow =SxSy =xy \neg \exists z =0Sz$, then $\vDash_B \gamma$.