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.
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 ni 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 \( (f_{0},f_{1},...,f_{{d-1}}) \) 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:
\( |B|\geq {\binom {n_{i}}{i-r}}+{\binom {n_{{i-1}}}{i-r-1}}+\ldots +{\binom {n_{j}}{j-r}}. \)
Lovász' simplified formulation
The following weaker but useful form is due to László Lovász (1993) Let A be a set of 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. If \( {\displaystyle |A|={\binom {x}{i}}} \) then \( {\displaystyle |B|\geq {\binom {x}{i-r}}}. \)
In this formulation, x need not be an integer. The value of the binomial expression is \( {\displaystyle {\binom {x}{i}}={\frac {x(x-1)\dots (x-i+1)}{i!}}}. \)
Ingredients of the proof
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
\( 123,124,134,234,125,135,235,145,245,345,\ldots . \)
Given a vector \( f=(f_{0},f_{1},...,f_{{d-1}}) \) with positive integer components, let Δf be the subset of the power set 2N 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:
Vector f is the f-vector of a simplicial complex Δ.
Δf is a simplicial complex.
\) f_{{i}}\leq f_{{i-1}}^{{(i)}},\quad 1\leq i\leq d-1. \)
The difficult implication is 1 ⇒ 2.
History
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.
See also
Sperner's theorem
References
Clements, G. F.; Lindström, B. (1969), "A generalization of a combinatorial theorem of Macaulay", Journal of Combinatorial Theory, 7: 230–238, doi:10.1016/S0021-9800(69)80016-5, MR 0246781. Reprinted in Gessel, Ira; Rota, Gian-Carlo, eds. (1987), Classic Papers in Combinatorics, Boston, Massachusetts: Birkhäuser Boston, Inc., pp. 416–424, doi:10.1007/978-0-8176-4842-8, ISBN 0-8176-3364-2, MR 0904286
Harper, L. H. (1966), "Optimal numberings and isoperimetric problems on graphs", Journal of Combinatorial Theory, 1: 385–393, doi:10.1016/S0021-9800(66)80059-5, MR 0200192
Katona, Gyula O. H. (1968), "A theorem of finite sets", in Erdős, Paul; Katona, Gyula O. H. (eds.), Theory of Graphs, Akadémiai Kiadó and Academic Press. Reprinted in Gessel & Rota (1987, pp. 381–401).
Knuth, Donald (2011), "7.2.1.3", The Art of Computer Programming, volume 4A: Combinatorial algorithms, part 1, p. 373.
Kruskal, Joseph B. (1963), "The number of simplices in a complex", in Bellman, Richard E. (ed.), Mathematical Optimization Techniques, University of California Press.
Lovász, László (1993), Combinatorial problems and exercises, Amsterdam: North-Holland.
Le, Dinh Van; Römer, Tim (2019), A Kruskal–Katona type result and applications, arXiv:1903.02998
Stanley, Richard (1996), Combinatorics and commutative algebra, Progress in Mathematics, 41 (2nd ed.), Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-3836-9.
Schützenberger, M.P. (1959), "A Characteristic Property of Certain Polynomials of E. F. Moore and C. E. Shannon", RLE Quarterly Progress Report, 55 (Processing and Transmission of Information): 117–118, retrieved 19 March 2019.
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