On the number of different variables required to define the n-density or the bounded n-width of Kripke frames with some consequences for Sahlqvist formulae

We show that both the $n$-density and the bounded $n$-width of Kripke frames can be modally defined not only with natural and well-known Sahlqvist formulae containing a linear number of different propositional variables but also with formulae of polynomial length with a logarithmic number of different propositional variables and then we prove that this exponential decrease in the number of variables leads us outside the class of Sahlqvist formulae.

Automated summary

The study demonstrates that the $n$-density and bounded $n$-width of Kripke frames can be defined using Sahlqvist formulae with either a linear number of propositional variables or with polynomial length formulae containing a logarithmic number of variables. However, it is shown that this exponential decrease in the number of variables takes the definition outside the class of Sahlqvist formulae.