100,061 research outputs found
On solution-free sets of integers
Given a linear equation , a set is
-free if does not contain any `non-trivial' solutions to
. We determine the precise size of the largest -free
subset of for several general classes of linear equations
of the form for fixed where .
Further, for all such linear equations , we give an upper bound on
the number of maximal -free subsets of . In the case when
and this bound is exact up to an error term in the exponent.
We make use of container and removal lemmas of Green to prove this result.
Our results also extend to various linear equations with more than three
variables.Comment: 14 pages, to appear in Acta Arithmetic
On the complexity of finding and counting solution-free sets of integers
Given a linear equation , a set of integers is
-free if does not contain any `non-trivial' solutions to
. This notion incorporates many central topics in combinatorial
number theory such as sum-free and progression-free sets. In this paper we
initiate the study of (parameterised) complexity questions involving
-free sets of integers. The main questions we consider involve
deciding whether a finite set of integers has an -free subset
of a given size, and counting all such -free subsets. We also
raise a number of open problems.Comment: 27 page
Threshold functions and Poisson convergence for systems of equations in random sets
We present a unified framework to study threshold functions for the existence
of solutions to linear systems of equations in random sets which includes
arithmetic progressions, sum-free sets, -sets and Hilbert cubes. In
particular, we show that there exists a threshold function for the property
" contains a non-trivial solution of
", where is a random set and each of
its elements is chosen independently with the same probability from the
interval of integers . Our study contains a formal definition of
trivial solutions for any combinatorial structure, extending a previous
definition by Ruzsa when dealing with a single equation.
Furthermore, we study the behaviour of the distribution of the number of
non-trivial solutions at the threshold scale. We show that it converges to a
Poisson distribution whose parameter depends on the volumes of certain convex
polytopes arising from the linear system under study as well as the symmetry
inherent in the structures, which we formally define and characterize.Comment: New version with minor corrections and changes in notation. 24 Page
Independence and counting problems in Combinatorics and Number theory
The method ofhypergraph containers has become a very important tool for dealing with problems which can be phrased in the language of independent sets in hypergraphs. This method has applications to numerous problems in combinatorics and other areas. In this thesis we consider examples of such problems; in particular problems concerning sets avoiding solutions to a given system of linear equations L (known as L-free sets) or graphs avoiding copies of a given graph H (H-free graphs). First we attack a number of questions relating to L-free sets. For example, we give various bounds on the number of maximal L-free subsets of [n] for three-variable homogeneous linear equations L. We then use containers to prove results corresponding to problems concerning tuples of disjoint independentsets in hypergraphs. In particular we generalise the random Ramsey theorem of Rodl and Rucinski by providing a resilience analogue. We obtain similar results for L-free sets. Finally we consider the Maker-Breaker game where Maker's aim is to obtain a solution to a given system of linear equations L amongst a random set of integers. We determine the threshold probability for this game for a large class of systems L
On equations over sets of integers
Systems of equations with sets of integers as unknowns are considered. It is
shown that the class of sets representable by unique solutions of equations
using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in
T} and with ultimately periodic constants is exactly the class of
hyper-arithmetical sets. Equations using addition only can represent every
hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can
also be represented by equations over sets of natural numbers equipped with
union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in
T, \: m \geqslant n}. Testing whether a given system has a solution is
-complete for each model. These results, in particular, settle the
expressive power of the most general types of language equations, as well as
equations over subsets of free groups.Comment: 12 apges, 0 figure
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