Defining long words succinctly in FO and MSO

We consider the length of the longest word definable in FO and MSO via a formula of size n. For both logics we obtain as an upper bound for this number an exponential tower of height linear in n. We prove this by counting types with respect to a fixed quantifier rank. As lower bounds we obtain for both FO and MSO an exponential tower of height in the order of a rational power of n. We show these lower bounds by giving concrete formulas defining word representations of levels of the cumulative hierarchy of sets. For the two-variable fragment of FO we obtain quadratic lower and upper bounds for the definability numbers of quantifier rank k fragments. In addition, we consider the Löwenheim–Skolem and Hanf numbers of these logics on words and obtain similar bounds for these as well.