Erdős–Ko–Rado theorem

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, ${\displaystyle r}$, and are all subsets of some larger set of size ${\displaystyle n}$. 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 ${\displaystyle n}$ is large enough for the problem to be nontrivial (${\displaystyle n\geq 2r}$) this construction produces the largest possible intersecting families. When ${\displaystyle n=2r}$ there are other equally-large families, but for larger values of ${\displaystyle n}$ 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.