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Almost-Fisher families

A classic theorem in combinatorial design theory is Fisher's inequality, which states that a family $\mathcal F$ of subsets of $[n]$ with all pairwise intersections of size $λ$ can have at most $n$ non-empty sets. One may weaken the condition by requiring that for every set in $\mathcal F$, all but at most $k$ of its pairwise intersections have size $λ$. We call such families $k$-almost $λ$-Fisher. Vu was the first to study the maximum size of such families, proving that for $k=1$ the largest family has $2n-2$ sets, and characterising when equality is attained. We substantially refine his result, showing how the size of the maximum family depends on $λ$. In particular we prove that for small $λ$ one essentially recovers Fisher's bound. We also solve the next open case of $k=2$ and obtain the first non-trivial upper bound for general $k$.