  We generalize and extend the ideas in a recent paper of Chiarelli, Hatami and
Saks to prove new bounds on the number of relevant variables for boolean
functions in terms of a variety of complexity measures. Our approach unifies
and refines all previously known bounds of this type. We also improve Nisan and
Szegedy's well-known block sensitivity vs. degree inequality by a constant
factor, thereby improving Huang's recent proof of the sensitivity conjecture by
the same constant.
