Irreducible hypergraphs for Hall-type conditions, and arc-minimal digraph expanders

Suppose that a hypergraph H=(V,E) satisfies a Hall-type condition of the form |⋃F|⩾r|F|+δ whenever 0̸≠F⊆E, but that this condition fails if any vertex (element) is removed from any edge (set) in E. How large an edge can H contain? It is proved here that there is no upper bound to the size of an edge if r is irrational, but that if r=p/q as a rational in its lowest terms then H can have no edge with more than max{p,p+⌈δ⌉} vertices (and if δ<0 then H must have an edge with at most ⌈(p−1)/q⌉ vertices). If δ⩽0 then the upper bound p is sharp, but if δ>0 then the bound p+⌈δ⌉ can be improved in some cases (we conjecture, in most cases). As a generalization of this problem, suppose that a digraph D=(V,A) satisfies an expansion condition of the form |N+(X)∖X|⩾r|X|+δ whenever 0̸≠X⊆S, where S is a fixed subset of V, but that this condition fails if any arc is removed from D. It is proved that if r=p/q as a rational in its lowest terms, then every vertex of S has outdegree at most max{p+q,p+q+⌈δ⌉−1}, and at most max{p,p+⌈δ⌉} if S is independent, but that if r is irrational then the vertices of S can have arbitrarily large outdegree.