Erdős–Ko–Rado theorem

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In combinatorics, the Erdős–Ko–Rado theorem of Paul Erdős, Chao Ko, and Richard Rado is a theorem on hypergraphs, specifically, on uniform hypergraphs of rank r.

The theorem is as follows. If , and A is a family of distinct subsets of {1,2,...,n}, such that each subset is of size r (thus making A a uniform hypergraph of rank r), and each pair of subsets intersects, then the maximum number of sets that can be in A is given by the binomial coefficient

.

According to Erdős 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 r, 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 r can be increased to size r to make the above statement apply.

[edit] Proof

The original proof of 1961 used induction on n. In 1972, Gyula O. H. Katona gave the following short and beautiful proof using double counting.

Suppose we have some such set A. Arrange the elements of {1,2,...,n} in any cyclic order, and inquire how many sets from A can form intervals within this cyclic order. For example if n = 8 and r = 3, we could arrange the numbers 1,...,8 as

[3,1,5,4,2,7,6,8]

and intervals would be

{1,3,5},{1,4,5},{2,4,5},{2,4,7},{2,6,7},{6,7,8},{3,6,8},{1,3,8}.

(Key step) At most r of these intervals can be in A. To see this, note that if

(a₁,a₂,...,a_r)

is one of these intervals in A then for every i, i = 1,2,...,r − 1, there is at most one interval in A which separates a_i from a_{i + 1}, i.e. contains precisely one of a_i and a_{i + 1}. (If there were two, they would be disjoint, since .) Furthermore, if n > 2r and there are r intervals in A, then they must contain some element in common.

There are (n − 1)! cyclic orders, and each set from A is an interval in precisely r!(n − r)! of them. Therefore the average number of intervals that A has in a random cyclic order must be

Rearranging the inequality, we get

establishing the theorem.

[edit] Further reading

• Erdős, P. (1987). "My joint work with Richard Rado". Surveys in Combinatorics, London Math. Soc. Lecture Note Series 123: 53-80. 

• Erdős, P.; Ko, C. & Rado, R (1961), “Intersection theorems for systems of finite sets”, Quarterly Journal of Mathematics, Oxford Series, series 2 12: 313-320, <http://www.renyi.hu/~p_erdos/1978-12.pdf> 

• Katona, G. O. H. (1972). "A simple proof of the Erdös-Chao Ko-Rado theorem". Journal of Combinatorial Theory, Series B 13: 183-184. doi:10.1016/0095-8956(72)90054-8.