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The back-and-forth method (Ehrenfeucht-Fraïssé games) generalizes beyond finite games to prove elementary equivalence and construct automorphisms. The method works by alternately placing witnesses and showing that the player avoiding an error can always respond in the other structure, building an elementary map piece by piece.
From Ehrenfeucht-Fraïssé games, you know the core idea: Duplicator wins the n-round game on structures A and B if and only if A and B satisfy the same first-order sentences of quantifier rank ≤ n. In the finite game, Spoiler picks an element in one structure, Duplicator picks a matching element in the other, and Duplicator must maintain a partial isomorphism at every step. The back-and-forth method extends this to build a full elementary embedding or isomorphism when the game can be played for infinitely many rounds.
The key move is to extend back and forth alternately. In the infinite game, after all n rounds have been played, the current partial map f: {a₁,…,aₙ} → {b₁,…,bₙ} must be extendable in *both* directions — forward (Spoiler plays in A, Duplicator responds in B) *and* backward (Spoiler plays in B, Duplicator responds in A). If Duplicator can always respond in either direction, the union of all the partial maps built over infinitely many rounds gives a total map that is an isomorphism. This is exactly how Cantor proved that any two countable dense linear orders without endpoints are isomorphic: at each step, you can always find a suitable element between any two existing ones in a dense order, so the back-and-forth construction never gets stuck.
The advanced applications arise when you use the method to build elementary embeddings rather than full isomorphisms. Here, Duplicator's response must preserve all first-order formulas, not just atomic ones. A sufficient condition is that the two structures are ω-saturated and elementarily equivalent — then Duplicator can always respond using type-realization: every finite type realized in A is realized in B, so whatever partial elementary map you have can always be extended one step. This yields the theorem that any two countable ω-categorical structures with the same complete theory are isomorphic.
A subtle variant is the back-and-forth system, a collection of partial isomorphisms closed under extensions in both directions. If such a system is non-empty and the structures are countable, you can thread a total isomorphism through the system by a standard diagonalization: enumerate all elements, alternating between A and B, and at each step use the closure property to extend the current partial map. The back-and-forth method thereby converts a *local* condition (every finite partial map can be extended) into a *global* conclusion (a total isomorphism exists), making it one of the most powerful construction techniques in model theory.