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

Relationships between the number of inputs and other complexity measures of Boolean functions