Almost-Fisher families

TL;DR

This paper studies families of sets with near-constant pairwise intersection sizes, extending Fisher's inequality, and provides new bounds on their maximum sizes depending on parameters $k$ and $\lambda$.

Contribution

It refines Vu's results on $k$-almost $\lambda$-Fisher families, especially for small $\lambda$, and introduces bounds for the case $k=2$ and general $k$.

Findings

For small $\\lambda$, the maximum size approaches Fisher's original bound.

Solved the open case of $k=2$ for the maximum family size.

Provided the first non-trivial upper bounds for general $k$.

Abstract

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 $\lambda$ 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 $\lambda$. We call such families $k$-almost $\lambda$-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 $\lambda$. In particular we prove that for small $\lambda$ 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 $k$.