= Complete-intersection candidate family
{title2=$\mathcal A_i=\{A\in[n]^{(r)}:|A\cap[t+2i]|\geq t+i\}$}
= Frankl family
{c}
{synonym}
Two members of this <uniform set family> have at least $2(t+i)-(t+2i)=t$ common points in its distinguished prefix, so it is a <t-intersecting family>. Nonempty indices satisfy $0\leq i\leq r-t$. Consecutive-family size comparison reduces to the threshold $n_{i+1}=(r-t+1)(2+(t-1)/(i+1))$: gain sets contain both newly introduced coordinates and $t+i-1$ earlier points; loss sets contain neither and $t+i$ earlier points. Counting these sets with <binomial coefficients> proves increase below, equality at and decrease above this threshold.
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