Regular bipartite graphs and intersecting families

In this paper we present a simple unifying approach to prove several statements about intersecting and cross-intersecting families, including the Erd\H os–Ko–Rado theorem, the Hilton–Milner theorem, a theorem due to Frankl concerning the size of intersecting families with bounded maximal degree, and versions of results on the sum of sizes of non-empty cross-intersecting families due to Frankl and Tokushige. Several new stronger results are also obtained. Our approach is based on the use of regular bipartite graphs. These graphs are quite often used in Extremal Set Theory problems, however, the approach we develop proves to be particularly fruitful.