5,745 research outputs found
On Minimum Saturated Matrices
Motivated by the work of Anstee, Griggs, and Sali on forbidden submatrices
and the extremal sat-function for graphs, we introduce sat-type problems for
matrices. Let F be a family of k-row matrices. A matrix M is called
F-admissible if M contains no submatrix G\in F (as a row and column permutation
of G). A matrix M without repeated columns is F-saturated if M is F-admissible
but the addition of any column not present in M violates this property. In this
paper we consider the function sat(n,F) which is the minimum number of columns
of an F-saturated matrix with n rows. We establish the estimate
sat(n,F)=O(n^{k-1}) for any family F of k-row matrices and also compute the
sat-function for a few small forbidden matrices.Comment: 31 pages, included a C cod
Saturation in the Hypercube and Bootstrap Percolation
Let denote the hypercube of dimension . Given , a spanning
subgraph of is said to be -saturated if it does not
contain as a subgraph but adding any edge of
creates a copy of in . Answering a question of Johnson and Pinto, we
show that for every fixed the minimum number of edges in a
-saturated graph is .
We also study weak saturation, which is a form of bootstrap percolation. A
spanning subgraph of is said to be weakly -saturated if the
edges of can be added to one at a time so that each
added edge creates a new copy of . Answering another question of Johnson
and Pinto, we determine the minimum number of edges in a weakly
-saturated graph for all . More generally, we
determine the minimum number of edges in a subgraph of the -dimensional grid
which is weakly saturated with respect to `axis aligned' copies of a
smaller grid . We also study weak saturation of cycles in the grid.Comment: 21 pages, 2 figures. To appear in Combinatorics, Probability and
Computin
On Saturated -Sperner Systems
Given a set , a collection is said to
be -Sperner if it does not contain a chain of length under set
inclusion and it is saturated if it is maximal with respect to this property.
Gerbner et al. conjectured that, if is sufficiently large with respect to
, then the minimum size of a saturated -Sperner system
is . We disprove this conjecture
by showing that there exists such that for every and there exists a saturated -Sperner system
with cardinality at most
.
A collection is said to be an
oversaturated -Sperner system if, for every
, contains more
chains of length than . Gerbner et al. proved that, if
, then the smallest such collection contains between and
elements. We show that if ,
then the lower bound is best possible, up to a polynomial factor.Comment: 17 page
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