35,674 research outputs found
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
Four-term progression free sets with three-term progressions in all large subsets
This paper is mainly concerned with sets which do not contain four-term
arithmetic progressions, but are still very rich in three term arithmetic
progressions, in the sense that all sufficiently large subsets contain at least
one such progression. We prove that there exists a positive constant and a
set which does not contain a four-term arithmetic
progression, with the property that for every subset with , contains a nontrivial three term arithmetic progression.
We derive this from a more general quantitative Roth-type theorem in random
subsets of , which improves a result of
Kohayakawa-Luczak-R\"odl/Tao-Vu. We also discuss a similar phenomenon over the
integers, where we show that for all , and all sufficiently large
, there exists a four-term progression-free set of size
with the property that for every subset with contains a nontrivial three term
arithmetic progression. Finally, we include another application of our methods,
showing that for sets in or the property of
"having nontrivial three-term progressions in all large subsets" is almost
entirely uncorrelated with the property of "having large additive energy".Comment: minor updates including suggestions from referee
On sets of integers which contain no three terms in geometric progression
The problem of looking for subsets of the natural numbers which contain no
3-term arithmetic progressions has a rich history. Roth's theorem famously
shows that any such subset cannot have positive upper density. In contrast,
Rankin in 1960 suggested looking at subsets without three-term geometric
progressions, and constructed such a subset with density about 0.719. More
recently, several authors have found upper bounds for the upper density of such
sets. We significantly improve upon these bounds, and demonstrate a method of
constructing sets with a greater upper density than Rankin's set. This
construction is optimal in the sense that our method gives a way of effectively
computing the greatest possible upper density of a geometric-progression-free
set. We also show that geometric progressions in Z/nZ behave more like Roth's
theorem in that one cannot take any fixed positive proportion of the integers
modulo a sufficiently large value of n while avoiding geometric progressions.Comment: 16 page
Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions
Two well studied Ramsey-theoretic problems consider subsets of the natural
numbers which either contain no three elements in arithmetic progression, or in
geometric progression. We study generalizations of this problem, by varying the
kinds of progressions to be avoided and the metrics used to evaluate the
density of the resulting subsets. One can view a 3-term arithmetic progression
as a sequence , where , a nonzero
integer. Thus avoiding three-term arithmetic progressions is equivalent to
containing no three elements of the form with , the set of integer translations. One can similarly
construct related progressions using different families of functions. We
investigate several such families, including geometric progressions ( with a natural number) and exponential progressions ().
Progression-free sets are often constructed "greedily," including every
number so long as it is not in progression with any of the previous elements.
Rankin characterized the greedy geometric-progression-free set in terms of the
greedy arithmetic set. We characterize the greedy exponential set and prove
that it has asymptotic density 1, and then discuss how the optimality of the
greedy set depends on the family of functions used to define progressions.
Traditionally, the size of a progression-free set is measured using the (upper)
asymptotic density, however we consider several different notions of density,
including the uniform and exponential densities.Comment: Version 1.0, 13 page
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