431 research outputs found
Forbidding intersection patterns between layers of the cube
A family is said to be an antichain
if for all distinct . A classic result
of Sperner shows that such families satisfy , which is easily seen to be best possible. One can
view the antichain condition as a restriction on the intersection sizes between
sets in different layers of . More generally one can ask,
given a collection of intersection restrictions between the layers, how large
can families respecting these restrictions be? Answering a question of Kalai,
we show that for most collections of such restrictions, layered families are
asymptotically largest. This extends results of Leader and the author.Comment: 16 page
Decomposing 1-Sperner hypergraphs
A hypergraph is Sperner if no hyperedge contains another one. A Sperner
hypergraph is equilizable (resp., threshold) if the characteristic vectors of
its hyperedges are the (minimal) binary solutions to a linear equation (resp.,
inequality) with positive coefficients. These combinatorial notions have many
applications and are motivated by the theory of Boolean functions and integer
programming. We introduce in this paper the class of -Sperner hypergraphs,
defined by the property that for every two hyperedges the smallest of their two
set differences is of size one. We characterize this class of Sperner
hypergraphs by a decomposition theorem and derive several consequences from it.
In particular, we obtain bounds on the size of -Sperner hypergraphs and
their transversal hypergraphs, show that the characteristic vectors of the
hyperedges are linearly independent over the reals, and prove that -Sperner
hypergraphs are both threshold and equilizable. The study of -Sperner
hypergraphs is motivated also by their applications in graph theory, which we
present in a companion paper
A group-theoretic approach to fast matrix multiplication
We develop a new, group-theoretic approach to bounding the exponent of matrix
multiplication. There are two components to this approach: (1) identifying
groups G that admit a certain type of embedding of matrix multiplication into
the group algebra C[G], and (2) controlling the dimensions of the irreducible
representations of such groups. We present machinery and examples to support
(1), including a proof that certain families of groups of order n^(2 + o(1))
support n-by-n matrix multiplication, a necessary condition for the approach to
yield exponent 2. Although we cannot yet completely achieve both (1) and (2),
we hope that it may be possible, and we suggest potential routes to that result
using the constructions in this paper.Comment: 12 pages, 1 figure, only updates from previous version are page
numbers and copyright informatio
The structure of the consecutive pattern poset
The consecutive pattern poset is the infinite partially ordered set of all
permutations where if has a subsequence of adjacent
entries in the same relative order as the entries of . We study the
structure of the intervals in this poset from topological, poset-theoretic, and
enumerative perspectives. In particular, we prove that all intervals are
rank-unimodal and strongly Sperner, and we characterize disconnected and
shellable intervals. We also show that most intervals are not shellable and
have M\"obius function equal to zero.Comment: 29 pages, 7 figures. To appear in IMR
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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