A note on fractional disjoint transversals in hypergraphs

A transversal in a hypergraph H is a subset of vertices that has a nonempty intersection with every edge of H. A transversal family F of H is a family of (not necessarily distinct) transversals of H. The effective transversal-ratio of the family F is the ratio of the number of sets in F over the maximum times rF any element appears in F. The fractional disjoint transversal number FDT(H) is the supremum of the effective transversal-ratio taken over all transversal families. That is, FDT(H)=supF|F|∕rF. Using a connection with not-all-equal 3-SAT, we prove that if H is a 3-regular 3-uniform hypergraph, then FDT(H)≥2, which proves a known conjecture. Using probabilistic arguments, we prove that for all k≥3, if H is a k-regular k-uniform hypergraph, then FDT(H)≥1∕(1−(k−1k)1k1k−1), and that this bound is essentially best possible.