 Constant t-wise intersection dichotomy

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.