95 research outputs found
Infinite Monochromatic Sumsets for Colourings of the Reals.
N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of R so that no infinite sumset X + X is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any finite colouring c of R there is an infinite X ⊆ R so that c ↾ X + X is constant
The number of subsets of integers with no -term arithmetic progression
Addressing a question of Cameron and Erd\Ho s, we show that, for infinitely
many values of , the number of subsets of that do not
contain a -term arithmetic progression is at most , where
is the maximum cardinality of a subset of without
a -term arithmetic progression. This bound is optimal up to a constant
factor in the exponent. For all values of , we prove a weaker bound, which
is nevertheless sufficient to transfer the current best upper bound on
to the sparse random setting. To achieve these bounds, we establish a new
supersaturation result, which roughly states that sets of size
contain superlinearly many -term arithmetic progressions.
For integers and , Erd\Ho s asked whether there is a set of integers
with no -term arithmetic progression, but such that any -coloring
of yields a monochromatic -term arithmetic progression. Ne\v{s}et\v{r}il
and R\"odl, and independently Spencer, answered this question affirmatively. We
show the following density version: for every and , there
exists a reasonably dense subset of primes with no -term arithmetic
progression, yet every of size contains a
-term arithmetic progression.
Our proof uses the hypergraph container method, which has proven to be a very
powerful tool in extremal combinatorics. The idea behind the container method
is to have a small certificate set to describe a large independent set. We give
two further applications in the appendix using this idea.Comment: To appear in International Mathematics Research Notices. This is a
longer version than the journal version, containing two additional minor
applications of the container metho
Enumeration of three term arithmetic progressions in fixed density sets
Additive combinatorics is built around the famous theorem by Szemer\'edi
which asserts existence of arithmetic progressions of any length among the
integers. There exist several different proofs of the theorem based on very
different techniques. Szemer\'edi's theorem is an existence statement, whereas
the ultimate goal in combinatorics is always to make enumeration statements. In
this article we develop new methods based on real algebraic geometry to obtain
several quantitative statements on the number of arithmetic progressions in
fixed density sets. We further discuss the possibility of a generalization of
Szemer\'edi's theorem using methods from real algebraic geometry.Comment: 62 pages. Update v2: Corrected some references. Update v3:
Incorporated feedbac
Problems on infinite sumset configurations in the integers and beyond
In contrast to finite arithmetic configurations, relatively little is known
about which infinite patterns can be found in every set of natural numbers with
positive density. Building on recent advances showing infinite sumsets can be
found, we explore numerous open problems and obstructions to finding other
infinite configurations in every set of natural numbers with positive density.Comment: 37 page
Partition regularity of a system of De and Hindman
We prove that a certain matrix, which is not image partition regular over R
near zero, is image partition regular over N. This answers a question of De and
Hindman.Comment: 7 page
An improved lower bound for Folkman's theorem
Folkman's Theorem asserts that for each , there exists a
natural number such that whenever the elements of are
two-coloured, there exists a set of size with the property
that all the sums of the form , where is a nonempty
subset of , are contained in and have the same colour. In 1989,
Erd\H{o}s and Spencer showed that , where is
an absolute constant; here, we improve this bound significantly by showing that
for all .Comment: 5 pages, Bulletin of the LM
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