Almost-Fisher families

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Almost-Fisher families
A classic theorem in combinatorial design theory is Fisher’s inequality, which states that a family 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 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.
Shagnik Das, Benny Sudakov, Pedro Vieira
Added 06 Apr 2016
Updated 06 Apr 2016
Type Journal
Year 2016
Where JCT
Authors Shagnik Das, Benny Sudakov, Pedro Vieira
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