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Concept
Erdős–Ko–Rado theorem
Summary
In mathematics, the Erdős–Ko–Rado theorem limits the number of sets in a family of sets for which every two sets have at least one element in common. Paul Erdős, Chao Ko, and Richard Rado proved the theorem in 1938, but did not publish it until 1961. It is part of the field of combinatorics, and one of the central results of extremal set theory.
The theorem applies to families of sets that all have the same size, , and are all subsets of some larger set of size . One way to construct a family of sets with these parameters, each two sharing an element, is to choose a single element to belong to all the subsets, and then form all of the subsets that contain the chosen element. The Erdős–Ko–Rado theorem states that when is large enough for the problem to be nontrivial () this construction produces the largest possible intersecting families. When there are other equally-large families, but for larger values of only the families constructed in this way can be largest.
The Erdős–Ko–Rado theorem can also be described in terms of hypergraphs or independent sets in Kneser graphs. Several analogous theorems apply to other kinds of mathematical object than sets, including linear subspaces, permutations, and strings. They again describe the largest possible intersecting families as being formed by choosing an element and forming the family of all objects that contain the chosen element.
Suppose that is a family of distinct -element subsets of an -element set with , and that each two subsets share at least one element. Then the theorem states that the number of sets in is at most the binomial coefficient
The requirement that is necessary for the problem to be nontrivial: when , all -element sets intersect, and the largest intersecting family consists of all -element sets, with size .
The same result can be formulated as part of the theory of hypergraphs. A family of sets may also be called a hypergraph, and when all the sets (which are called "hyperedges" in this context) are the same size , it is called an -uniform hypergraph.
Official source
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