In this paper we present a simple unifying approach to prove severalstatements about intersecting and cross-intersecting families, including theErd\H os--Ko--Rado theorem, the Hilton--Milner theorem, a theorem due to Franklconcerning the size of intersecting families with bounded maximal degree, andversions of results on the sum of sizes of non-empty cross-intersectingfamilies due to Frankl and Tokushige. Several new stronger results are alsoobtained. Our approach is based on the use of regular bipartite graphs. These graphsare quite often used in Extremal Set Theory problems, however, the approach wedevelop proves to be particularly fruitful.
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机译：在本文中，我们提出了一种简单的统一方法来证明关于相交和交叉相交的家庭的几种陈述，包括Erd \ H os-Ko-Rado定理，Hilton-Milner定理，这是由于Frankl考虑到相交家庭的大小而定理的具有最大极限度，并且由于Frankl和Tokushige导致非空交叉相交族的大小之和的结果转换。还获得了几个新的更强的结果。我们的方法基于正则二部图的使用。这些图经常用在极端集理论问题中，但是，我们开发的方法被证明是特别有效的。
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