 Constant-intersection family bound

If distinct members of a set family on $[n]$ have intersection size $\lambda$, then its size is at most $n$ for $\lambda >0$, and at most $n+1$ for $\lambda =0$, assuming $n\ge 1$. In the positive case, use the Gram matrix of the characteristic vectors of sets; a member of size $\lambda$ is handled by disjoint remainders. In the zero case nonempty members are disjoint.