395 research outputs found
Extremal families for the Kruskal--Katona theorem
Given a family of --subsets of , its lower shadow is
the family of --subsets which are contained in at least one set in .
The celebrated Kruskal--Katona theorem gives the minimum cardinality of
in terms of the cardinality of . F\"uredi and Griggs (and
M\"ors) showed that the extremal families for this shadow minimization problem
in the Boolean lattice are unique for some cardinalities and asked for a
general characterization of these extremal families.
In this paper we prove a new combinatorial inequality from which yet another
simple proof of the Kruskal--Katona theorem can be derived. The inequality can
be used to obtain a characterization of the extremal families for this
minimization problem, giving an answer to the question of F\"uredi and Griggs.
Some known and new additional properties of extremal families can also be
easily derived from the inequality
f-vectors implying vertex decomposability
We prove that if a pure simplicial complex of dimension d with n facets has
the least possible number of (d-1)-dimensional faces among all complexes with n
faces of dimension d, then it is vertex decomposable. This answers a question
of J. Herzog and T. Hibi. In fact we prove a generalization of their theorem
using combinatorial methods
How to construct a flag complex with a given face vector
A method that often works for constructing a flag complex with a specified
face vector is given. This method can also be adapted to construct a
vertex-decomposable (and hence Cohen-Macaulay) flag complex with a specified
h-vector
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