145 research outputs found
Forbidden subposet problems for traces of set families
In this paper we introduce a problem that bridges forbidden subposet and
forbidden subconfiguration problems. The sets form a
copy of a poset , if there exists a bijection such that for any the relation implies
. A family of sets is \textit{-free} if
it does not contain any copy of . The trace of a family on a
set is .
We introduce the following notions: is
-trace -free if for any -subset , the family
is -free and is trace -free if it is
-trace -free for all . As the first instances of these problems
we determine the maximum size of trace -free families, where is the
butterfly poset on four elements with and determine the
asymptotics of the maximum size of -trace -free families for
. We also propose a generalization of the main conjecture of the area of
forbidden subposet problems
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
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Pre-compact families of finite sets of integers and weakly null sequences in Banach spaces
Two applications of Nash-Williams' theory of barriers to sequences on Banach
spaces are presented: The first one is the -saturation of ,
countable compacta. The second one is the construction of weakly-null sequences
generalizing the example of Maurey-Rosenthal
Two-part set systems
The two part Sperner theorem of Katona and Kleitman states that if is an
-element set with partition , and \cF is a family of subsets
of such that no two sets A, B \in \cF satisfy (or ) and for some , then |\cF| \le {n
\choose \lfloor n/2 \rfloor}. We consider variations of this problem by
replacing the Sperner property with the intersection property and considering
families that satisfiy various combinations of these properties on one or both
parts , . Along the way, we prove the following new result which may
be of independent interest: let \cF, \cG be families of subsets of an
-element set such that \cF and \cG are both intersecting and
cross-Sperner, meaning that if A \in \cF and B \in \cG, then and . Then |\cF| +|\cG| < 2^{n-1} and there are
exponentially many examples showing that this bound is tight
On -close Sperner systems
For a set of positive integers, a set system is said to be -close Sperner, if for any pair of distinct
sets in the skew distance belongs to . We reprove an extremal result of Boros,
Gurvich, and Milani\v c on the maximum size of -close Sperner set systems
for and generalize to and obtain slightly weaker bounds for
arbitrary . We also consider the problem when might include 0 and
reprove a theorem of Frankl, F\"uredi, and Pach on the size of largest set
systems with all skew distances belonging to
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