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In extremal set theory our usual goal is to find the maximal size of a familyof subsets of an $n$-element set satisfying a condition. A condition is calledchain-dependent, if it is satisfied for a family if and only if it is satisfiedfor its intersections with the $n!$ full chains. We introduce a method tohandle problems with such conditions, then show how it can be used to provethree classic theorems. Then, a theorem about families containing no two setssuch that $A\subset B$ and $\lambda \cdot |A| \le |B|$ is proved. Finally, weinvestigate problems where instead of the size of the family, the number of$\ell$-chains is maximized. Our method is to define a weight function on thesets (or $\ell$-chains) and use it in a double counting argument involving fullchains.

Chain-dependent Conditions in Extremal Set Theory