An asymptotically tight bound on countermodels for Łukasiewicz logic

Let ϕ be a formula of Łukasiewicz infinite-valued propositional logic having a total of l many occurrences of n distinct propositional variables (call l the length of ϕ). Results in Aguzzoli and Ciabattoni [Finiteness in infinite-valued Łukasiewicz logic, Journal of Logic, Language and Information, 9 (2000) 5–29] show that if ϕ is not a tautology then there is an MV chain A of cardinality ⩽ ⌊(l/n)n⌋ + 1 together with an evaluation eA of propositional variables in A, such that eA is a countermodel for ϕ, that is eA(ϕ)<1A. We show that for each integer n > 0 the function b(n, l) = (l/n)n + 1 yields an asymptotically tight upper bound on the maximum cardinality of the smallest MV algebras having countermodels for formulas of length l.