There's a large body of work on variations of the secretary problem. The one I know of which is most relevant to your question is section 4 of the following:
http://www.cs.cornell.edu/~rdk/papers/secArt4.pdf
Section 6 is particularly interesting, where you're further restricted - you want to choose multiple secretaries, but there's certain constraints on the sets you can choose. For example, the secretaries might be edges in a graph, and you can't pick a subset of edges which would result in a cycle (this is a "graphic matroid," which is an example of a mathematical object known as a matroid). The reason why this formulation of the problem is interesting is because the best known algorithm is O(sqrt(log k))-competitive (where k is the rank of the matroid), whereas it is conjectured that an O(1)-competitive algorithm exists.
Section 6 is particularly interesting, where you're further restricted - you want to choose multiple secretaries, but there's certain constraints on the sets you can choose. For example, the secretaries might be edges in a graph, and you can't pick a subset of edges which would result in a cycle (this is a "graphic matroid," which is an example of a mathematical object known as a matroid). The reason why this formulation of the problem is interesting is because the best known algorithm is O(sqrt(log k))-competitive (where k is the rank of the matroid), whereas it is conjectured that an O(1)-competitive algorithm exists.