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 Erdős–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. 1 Intersecting families Let [n] := {1, . . . , n} denote the standard n-element set. The family of all k-element subsets of [n] we denote by ( [n] k ) , and the set of all subsets of [n] we denote by 2. Any subset of 2 we call a family. We say that a family is intersecting, if any two sets from the family intersect. Probably, the first theorem, devoted to intersecting families, was the famous theorem of Erdős, Ko and Rado: Theorem A (Erdős, Ko, Rado, [1]). Let n ≥ 2k > 0. Then for any intersecting family F ⊂ ( [n] k ) one has |F| ≤ ( n−1 k−1 ) . It is easy to give an example of an intersecting family, on which the bound from Theorem A is attained: take the family of all k-element sets containing element 1. Any family, in which all sets contain a fixed element, we call trivially intersecting. What size can an intersecting family have, provided that it is not trivially (nontrivially) intersecting? For n = 2k it is easy to construct many intersecting families of size ( 2k−1 k−1 ) by choosing exactly one k-set out of each two complementary sets. For n > 2k the answer is given by the Hilton-Milner theorem. Theorem B (Hilton, Milner, [9]). Let n > 2k and F ⊂ ( [n] k ) be a nontrivially intersecting family. Then |F| ≤ ( n−1 k−1 )