On hierarchically closed fractional intersecting families

For a set L of positive proper fractions and a positive integer r >= 2, a fractional r-closed L-intersecting family is a collection F subset of P([n]) with the property that for any 2 <= t <= r and A(1),...,A(t) is an element of F there exists theta is an element of L such that |A(1) boolean AND center dot center dot center dot boolean AND A(t)| is an element of {theta|A(1)|,center dot center dot center dot, theta|A(t)|}. In this paper we show that for r >= 3 and L = {theta} any fractional r-closed theta-intersecting family has size at most linear in n, and this is best possible up to a constant factor. We also show that in the case theta = 1/2 we have a tight upper bound of [3n/2]-2 and that a maximal r-closed (1/2)-intersecting family is determined uniquely up to isomorphism.