 Complement-of-singleton extremizers

The sets $A_i=[m]\setminus \{i\}$ have every $t$-fold intersection of size $m-t$ and every $(t-2)$-fold intersection of size $m-t+2$. For $m\ge t+1$, their common intersection is empty and they attain $m=k+t-2$ in the constant t-wise intersection dichotomy.