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In combinatorics, the Erdős–Ko–Rado theorem of Paul Erdős, Chao Ko, and Richard Rado is a theorem on intersecting set families. It is part of the theory of hypergraphs, specifically, uniform hypergraphs of rank ''r''. The theorem is as follows. If and is a family of distinct subsets of such that each subset is of size and each pair of subsets intersects, then the maximum number of sets that can be in is given by the binomial coefficient : (Since a family of sets may be called a hypergraph, and since every set in has size ''r'', is a uniform hypergraph of rank ''r''.) According to the theorem was proved in 1938, but was not published until 1961 in an apparently more general form. The subsets in question were only required to be size ''at most'' , and with the additional requirement that no subset be contained in any other. This statement is not actually more general: any subset that has size less than can be increased to size to make the above statement apply. ==Proof== The original proof of 1961 used induction on ''n''. In 1972, Gyula O. H. Katona gave the following short proof using double counting. Suppose we have some such family of subsets ''A''. Arrange the elements of in any cyclic order, and consider the sets from ''A'' that form intervals of length ''r'' within this cyclic order. For example if ''n'' = 8 and ''r'' = 3, we could arrange the numbers into the cyclic order (3,1,5,4,2,7,6,8), which has eight intervals: :(3,1,5), (1,5,4), (5,4,2), (4,2,7), (2,7,6), (7,6,8), (6,8,3), and (8,3,1). However, it is not possible for all of the intervals of the cyclic order to belong to ''A'', because some pairs of them are disjoint. Katona's key observation is that at most ''r'' of the intervals for a single cyclic order may belong to ''A''. To see this, note that if (''a''1, ''a''2, ..., ''a''''r'') is one of these intervals in ''A'', then every other interval of the same cyclic order that belongs to ''A'' separates ''ai'' and ''a''''i'' + 1 for some ''i'' (that is, it contains precisely one of these two elements). The two intervals that separate these elements are disjoint, so at most one of them can belong to ''A''. Thus, the number of intervals in ''A'' is one plus the number of separated pairs, which is at most ''(r-1)''. Based on this idea, we may count the number of pairs (''S'',''C''), where ''S'' is a set in ''A'' and ''C'' is a cyclic order for which ''S'' is an interval, in two ways. First, for each set ''S'' one may generate ''C'' by choosing one of ''r''! permutations of ''S'' and (''n'' − ''r'')! permutations of the remaining elements, showing that the number of pairs is |''A''|''r''!(''n'' − ''r'')!. And second, there are (''n'' − 1)! cyclic orders, each of which has at most ''r'' intervals of ''A'', so the number of pairs is at most r(''n'' − 1)!. Combining these two counts gives the inequality : and dividing both sides by ''r''!(''n'' − ''r'')! gives the result : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Erdős–Ko–Rado theorem」の詳細全文を読む スポンサード リンク
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