= Constant t-wise intersection dichotomy
{title2=$m\leq k+t-2$}
Let $m\geq t\geq3$ distinct sets have every $t$-fold intersection of size $\lambda$. Either all contain a common set of size $\lambda$, or $m\leq k+t-2$, 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 $m\geq t$ excludes vacuous counterexamples.
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