i have tried to prove that every satisfiable sentence with at most two variables has a finite model, but my attempts were unsuccessful... this is an exercise from the wider model thoery book written by Hodges (ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS, Volume 42 - model theory, page 111). It follows an exercise about Immerman's pebble game, probably as an application. it's easy to see that proving the following problem will suffice: given a structure A, prove that for every number n, there is a finite structure B such that player II has a winning strategy in immerman's pebble game of length n with 2 pebbles.