In algebraic combinatorics, the Kruskal–Katona theorem gives a complete characterization of the f-vectors of abstract simplicial complexes.
It includes as a special case the Erdős–Ko–Rado theorem and can be restated in terms of uniform hypergraphs.
It is named after Joseph Kruskal and Gyula O. H. Katona, but has been independently discovered by several others.
Given two positive integers N and i, there is a unique way to expand N as a sum of binomial coefficients as follows: This expansion can be constructed by applying the greedy algorithm: set ni to be the maximal n such that
Define An integral vector
-dimensional simplicial complex if and only if Let A be a set consisting of N distinct i-element subsets of a fixed set U ("the universe") and B be the set of all
-element subsets of the sets in A.
Then the cardinality of B is bounded below as follows: The following weaker but useful form is due to László Lovász (1993, 13.31b).
-element subsets of the sets in A.
In this formulation, x need not be an integer.
The value of the binomial expression is
For every positive i, list all i-element subsets a1 < a2 < … ai of the set N of natural numbers in the colexicographical order.
For example, for i = 3, the list begins Given a vector
with positive integer components, let Δf be the subset of the power set 2N consisting of the empty set together with the first
i-element subsets of N in the list for i = 1, …, d. Then the following conditions are equivalent: The difficult implication is 1 ⇒ 2.
The theorem is named after Joseph Kruskal and Gyula O. H. Katona, who published it in 1963 and 1968 respectively.
According to Le & Römer (2019), it was discovered independently by Kruskal (1963), Katona (1968), Marcel-Paul Schützenberger (1959), Harper (1966), and Clements & Lindström (1969).
Donald Knuth (2011) writes that the earliest of these references, by Schützenberger, has an incomplete proof.