 Generator replacement proof of the small-support intersection lemma

Let $m$ be the smallest supporting prefix for a maximum left-compressed set family of $k$-sets that is t-intersecting, and set $N=n-m$. The maximum-support generator fibre gives exclusive count $\left(\genfrac{}{}{0ex}{}{N}{k-a}\right)$ for a size-$a$ generator containing $m$. By tight pairs of left-compressed generators, only classes $a,b$ with $a+b=m+t$ can conflict after deleting $m$. For unequal such sizes, replacing both classes by the shortened generators of either class would implyWhen $n\ge 2k-t+2$ and the classes have positive fibres, the reverse inequality is strict. For the remaining central class $a=t+j$, $m=t+2j$, some earlier coordinate is absent from at least $j/(m-1)$ of its shortened generators. Those generators are mutually t-intersecting. Replacing the central class by that subfamily has gain-to-loss ratio at leastwhich exceeds one exactly when $n>(k-t+1)(2+(t-1)/j)$. This forces the support bound in the small-support generating lemma for extremal intersecting families.