Let $FO$ be first-order logic and $FO^k$ be $k$-variable segment of $FO$, i.e. $FO^k$ has only $k$ variables.


To my understanding, for every sentence $\varphi\in FO$ there exists a sentence $\psi\in FO^2$ such that for all finite structures $\mathfrak{A}$ with linear order it is the case that $\mathfrak{A}\vDash\varphi$ iff $\mathfrak{A}\vDash\psi$.


Is this true? Are there any assumptions about the vocabulary? 



$2$-variable segment of FO over ordered, finite structures
It is not the case. As an example, the following paper:


Kouck´y, M., Lautemann, C., Poloczek, S., Th´erien, D.: Circuit lower bounds via
Ehrenfeucht-Fraiss´e games, 2006.


shows that, over words, FO[+, $\times$] with 2 variables is equivalent to AC$^0$ with *linear size* circuits --- while FO[+, $\times$] is equivalent to the whole class AC$^0$.


Those two classes are known to differ (S. Chaudhuri and J. Radhakrishnan: Deterministic restrictions in circuit complexity, 1996).


