On the ratio of maximum and minimum degree in maximal intersecting families

To study how balanced or unbalanced a maximal intersecting family F⊆([n]r) is we consider the ratio R(F)=Δ(F)δ(F) of its maximum and minimum degree. We determine the order of magnitude of the function m(n,r)m(n,r), the minimum possible value of R(F)R(F), and establish some lower and upper bounds on the function M(n,r)M(n,r), the maximum possible value of R(F)R(F). To obtain constructions that show the bounds on m(n,r)m(n,r) we use a theorem of Blokhuis on the minimum size of a non-trivial blocking set in projective planes.