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 ) , the minimum possible value of R ( F ) , and establish some lower and upper bounds on the function M ( n , r ) , the maximum possible value of R ( F ) . To obtain constructions that show the bounds on m ( n , r ) we use a theorem of Blokhuis on the minimum size of a non-trivial blocking set in projective planes.