Abstract: Given a disjoint union of hypergraphs H_1,...,H_m on the same vertex set, an m-edge hypergraph F is a transversal if each of its edges comes from a distinct hypergraph H_i. How large does the minimum degree of each H_i need to be so that we can always find a F-transversal?
Each H_i in the collection could be identical, hence the minimum degree of each H_i needs to be large enough to be able to find a copy of F in H_i. Since its general introduction by Joos and Kim, a growing body of work has shown that in many cases, this lower bound is tight.
In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac-type results for (powers of) Hamilton cycles.
