Multi-Structural Games and Number of Quantifiers

We study multi-structural games, played on two sets \mathcal A and \mathcal B of structures. These games generalize Ehrenfeucht-Fraïssé games. Whereas Ehrenfeucht-Fraïssé games capture the quantifier rank of a first-order sentence, multi-structural games capture the number of quantifiers, in the sense that Spoiler wins the r-round game if and only if there is a first-order sentence φ with at most r quantifiers, where every structure in \mathcal A satisfies φ and no structure in \mathcal B satisfies φ. We use these games to give a complete characterization of the number of quantifiers required to distinguish linear orders of different sizes, and develop machinery for analyzing structures beyond linear orders.