Kruskal–Katona theorem

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. The theorem is named after Joseph Kruskal and Gyula O. H. Katona. It was independently proved by Marcel Schützenberger, but his contribution escaped notice for several years.

Statement

Given two positive integers N and i, there is a unique way to expand N as a sum of binomial coefficients as follows:

$N={\binom {n_{i}}{i}}+{\binom {n_{i-1}}{i-1}}+\ldots +{\binom {n_{j}}{j}},\quad n_{i}>n_{i-1}>\ldots >n_{j}\geq j\geq 1.$

This expansion can be constructed by applying the greedy algorithm: set n_i to be the maximal n such that $N\geq {\binom {n}{i}},$ replace N with the difference, i with i − 1, and repeat until the difference becomes zero. Define

$N^{(i)}={\binom {n_{i}}{i+1}}+{\binom {n_{i-1}}{i}}+\ldots +{\binom {n_{j}}{j+1}}.$

Statement for simplicial complexes

An integral vector (f₀, f₁, … f_d −1 ) is the f-vector of some (d −1 )-dimensional simplicial complex if and only if

$0\leq f_{i}\leq f_{i-1}^{(i)},\quad 1\leq i\leq d-1.$

Statement for uniform hypergraphs

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 (i −r )-element subsets of the sets in A. Expand N as above. Then the cardinality of B is bounded below as follows:

$|B|\geq {\binom {n_{i}}{i-r}}+{\binom {n_{i-1}}{i-r-1}}+\ldots +{\binom {n_{j}}{j-r}}.$

Suppose that U is the union of the sets in A and that C is the set of all (i + r)-element supersets of the sets in A. Then the cardinality of C is bounded above as follows:

$|C|\leq {\binom {n_{i}}{i+r}}+{\binom {n_{i-1}}{i+r-1}}+\ldots +{\binom {n_{j}}{j+r}}.$

Ingredients of the proof

For every positive i, list all i-element subsets a₁ < a₂ < … a_i of the set N of natural numbers in the reverse lexicographic order. For example, for i = 3, the list begins

$123,124,134,234,125,135,235,145,245,345,\ldots .$

Given a vector f = (f₀, f₁, …, f_d −1 ) with positive integer components, let Δ_f be the subset of the power set 2^N consisting of the empty set together with the first f_i − 1 i-element subsets of N in the list for i = 1, …, d. Then the following conditions are equivalent:

1. Vector f is the f-vector of a simplicial complex Δ.
2. Δ_f is a simplicial complex.
3. $f_{i}\leq f_{i-1}^{(i)},\quad 1\leq i\leq d-1.$

The difficult implication is 1 ⇒ 2.

References

• Kruskal, J. B. (1963), "The number of simplices in a complex", in Bellman, R. (ed.), Mathematical Optimization Techniques, University of California Press.

• Katona, G. O. H. (1968), "A theorem of finite sets", in Erdős, P.; Katona, G. O. H. (eds.), Theory of Graphs, Akadémiai Kiadó and Academic Press.

• Knuth, D., The Art of Computer Programming, pre-fascicle 3a: Generating all combinations {{citation}}: External link in |title= (help). Contains a proof via a more general theorem in discrete geometry.

• Stanley, Richard (1996), Combinatorics and commutative algebra, Progress in Mathematics, vol. 41 (2nd ed.), Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-3836-9.

External links

• Kruskal-Katona theorem on the polymath1 wiki