= Generating family for a uniform set family
{title2=$\mathcal F=\{A\in[n]^{(r)}:\exists G\in\mathcal G,\ G\subseteq A\}$}
= Generating families for a uniform set family
{synonym}
A generating family consists of sets of size at most $r$ whose $r$-element supersets are exactly the prescribed <uniform set family>. Small-support generators express dependence on a bounded prefix rather than on the full ground set. Taking the entire family itself as generators always works, but extremal intersection problems admit much smaller supporting intervals.
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