8 research outputs found
Almost-Fisher families
A classic theorem in combinatorial design theory is Fisher's inequality,
which states that a family of subsets of with all pairwise
intersections of size can have at most non-empty sets. One may
weaken the condition by requiring that for every set in , all but
at most of its pairwise intersections have size . We call such
families -almost -Fisher. Vu was the first to study the maximum
size of such families, proving that for the largest family has
sets, and characterising when equality is attained. We substantially refine his
result, showing how the size of the maximum family depends on . In
particular we prove that for small one essentially recovers Fisher's
bound. We also solve the next open case of and obtain the first
non-trivial upper bound for general .Comment: 27 pages (incluiding one appendix
Almost intersecting families
Let be integers, . Let be a
family of -subsets of~. The family is called intersecting
if for all . It is called
almost intersecting if it is not intersecting but to every
there is at most one satisfying .
Gerbner et al. proved that if then holds for almost intersecting families. The main result
implies the considerably stronger and best possible bound for