Erdős–Ko–Rado theorem

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 $ngeq2r$, 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

:$\{n-1\}\; choose\; \{r-1\}$.

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.

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

:$(a\_1,a\_2,...,a\_r)$

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 $ngeq2r$.) 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

:$frac\{|A|cdot\; r!(n-r)!\}\{(n-1)!\}leq\; r.$

Rearranging the inequality, we get

:$|A|leqfrac\{r(n-1)!\}\{r!(n-r)!\}=frac\{(n-1)!\}\{(r-1)!(n-r)!\}=\{n-1choose\; r-1\},$

establishing the theorem.

Further reading

* cite conference
first = P.
last = Erdős
authorlink = Paul Erdős
title = My joint work with Richard Rado
booktitle = Surveys in Combinatorics, London Math. Soc. Lecture Note Series
volume = 123
year = 1987
pages = 53-80
url = http://www.renyi.hu/~p_erdos/1987-12.pdf

* Citation
first1 = P.
last1 = Erdős
author1-link = Paul Erdős
first2 = C.
last2 = Ko
author2-link = Chao Ko
first3 = R
last3 = Rado
author3-link = Richard Rado
title = Intersection theorems for systems of finite sets
journal = Quarterly Journal of Mathematics, Oxford Series, series 2
volume = 12
year = 1961
pages = 313–320
url = http://www.renyi.hu/~p_erdos/1978-12.pdf
doi = 10.1093/qmath/12.1.313

* cite journal
first = G. O. H.
last = Katona
authorlink = Gyula O. H. Katona
title = A simple proof of the Erdös-Chao Ko-Rado theorem
journal = Journal of Combinatorial Theory, Series B
volume = 13
year = 1972
pages = 183–184
doi = 10.1016/0095-8956(72)90054-8