Lower Bounds on the Size of Maximum Independent Sets and Matchings in Hypergraphs of Rank Three

In this paper, we study lower bounds on the size of a maximum independent set and a maximum matching in a hypergraph of rank three. The rank in a hypergraph is the size of a maximum edge in the hypergraph. A hypergraph is simple if no two edges contain exactly the same vertices. Let H be a hypergraph and let a(H) and a'(H) be the size of a maximum independent set and a maximum matching, respectively, in H, where a set of vertices in H is independent (also called strongly independent in the literature) if no two vertices in the set belong to a common edge. Let H be a hypergraph of rank at most three and maximum degree at most three. We show that a(H)=|V(H)|/7 with equality if and only if H is the Fano plane. In fact, we show that if H is connected and different from the Fano plane, then a(H)=(3|V(H)|-1)/17 and we characterize the hypergraphs achieving equality in this bound. Using this result, we show that that if H is a simple connected 3-uniform hypergraph of order at least 8 and with maximum degree at most three, then a'(H)=3|E(H)|/17 and there is a connected 3-uniform hypergraph H of order 19 achieving this lower bound. Finally, we show that if H is a connected hypergraph of rank at most three that is not a complete hypergraph on 2?(H)+1 vertices, where ?(H) denotes the maximum degree in H, then a(H)=|V(H)|/2?(H) and this bound is asymptotically best possible. (c) 2012 Wiley Periodicals, Inc. J. Graph Theory (C) 2012 Wiley Periodicals, Inc. J. Graph Theory 72: 220-245, 2013