= Small-support generating lemma for extremal intersecting families
{title2=$n>(r-t+1)(2+(t-1)/(j+1))\Longrightarrow\mathcal G\subseteq\mathcal P([t+2j])$}
Assume $1\leq t\leq r\leq n$, $n>2r-t$, and $j\geq0$. A maximum-cardinality <left-compressed set family> of $r$-sets that is <t-intersecting> has a <generating family for a uniform set family> supported on $[t+2j]$ under the displayed strict inequality. Reflection gives a right-end supporting interval for a right-compressed optimum. The strict threshold is important: at a tie between consecutive candidate families, use the next larger support bound. This auxiliary generating result, combined with the <compatibility bound for complementary generating families>, supplies the upper bound in the <Ahlswede-Khachatrian theorem>.
Back to article page
