Constant t-wise intersection dichotomy ($m\le k+t-2$)

Let $m\ge t\ge 3$ distinct sets have every $t$-fold intersection of size $\lambda$. Either all contain a common set of size $\lambda$, or $m\le k+t-2$, where $k$ is the minimum size of a $(t-2)$-fold intersection. Fix such a minimum intersection and restrict the remaining sets to it. Equal traces give the common set; distinct traces satisfy the constant-intersection family bound. The qualification $m\ge t$ excludes vacuous counterexamples.

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.