Effectiveness of Hindman's theorem for bounded sums

We consider the strength and effective content of restricted versions of Hindman's Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let $\mathsf{HT}^{\leq n}_k$ denote the assertion that for each $k$-coloring $c$ of $\mathbb{N}$ there is an infinite set $X \subseteq \mathbb{N}$ such that all sums $\sum_{x \in F} x$ for $F \subseteq X$ and $0 < |F| \leq n$ have the same color. We prove that there is a computable $2$-coloring $c$ of $\mathbb{N}$ such that there is no infinite computable set $X$ such that all nonempty sums of at most $2$ elements of $X$ have the same color. It follows that $\mathsf{HT}^{\leq 2}_2$ is not provable in $\mathsf{RCA}_0$ and in fact we show that it implies $\mathsf{SRT}^2_2$ in $\mathsf{RCA}_0$. We also show that there is a computable instance of $\mathsf{HT}^{\leq 3}_3$ with all solutions computing $0'$. The proof of this result shows that $\mathsf{HT}^{\leq 3}_3$ implies $\mathsf{ACA}_0$ in $\mathsf{RCA}_0$

The scarcity of products in βS ∖ S

Let S be a discrete semigroup and let the Stone–Čech compactification βS of S have the operation extending that of S which makes βS a right topological semigroup with S contained in its topological center. Let S ⁎ = β S ∖ S . Algebraically, the set of products S ⁎ S ⁎ tends to be rather large, since it often contains the smallest ideal of βS. We establish here sufficient conditions involving mild cancellation assumptions and assumptions about the cardinality of S for S ⁎ S ⁎ to be topologically small, that is for S ⁎ S ⁎ to be nowhere dense in S ⁎ , or at least for S ⁎ ∖ S ⁎ S ⁎ to be dense in S ⁎ . And we provide examples showing that these conditions cannot be significantly weakened. These extend results previously known for countable semigroups. Other results deal with large sets missing S ⁎ S ⁎ whose elements have algebraic properties, such as being right cancelable and generating free semigroups in βS

Algebraic and topological equivalences in the Stone-Čech compactification of a discrete semigroup

AbstractWe consider the Stone-Čech compactification βS of a countably infinite discrete commutative semigroup S. We show that, under a certain condition satisfied by all cancellative semigroups S, the minimal right ideals of βS will belong to 2c homeomorphism classes. We also show that the maximal groups in a given minimal left ideal will belong to 2c homeomorphism classes. The subsets of βS of the form S + e, where e denotes an idempotent, will also belong to 2c homeomorphism classes.All the left ideals of βN of the form βN + e, where e denotes a nonminimal idempotent of βN, will be different as right topological semigroups. If e denotes a nonminimal idempotent of βZ, e + βZ will be topologically and algebraically isomorphic to precisely one other principal right ideal of βZ defined by an idempotent: −e + βZ. The corresponding statement for left ideals is also valid

Enumeration of intersecting families

AbstractIt is shown that the logarithm to the base 2 of the number of maximal intersecting families on m elements is asymptotically equal to (m−1n−1) where n = [12m]

Hypernatural Numbers as Ultrafilters

In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers

Pairwise sums in colourings of the reals

Suppose that we have a finite colouring of R. What sumset-type structures can we hope to find in some colour class? One of our aims is to show that there is such a colouring for which no uncountable set has all of its pairwise sums monochromatic. We also show that there is such a colouring such that there is no infinite set X with X + X (the pairwise sums from X, allowing repetition) monochromatic. These results assume CH. In the other direction, we show that if each colour class is measurable, or each colour class is Baire, then there is an infinite set X (and even an uncountable X, of size the reals) with X + X monochromatic. We also give versions for all of these results for k-wise sums in place of pairwise sums

Topological properties of some algebraically defined subsets of βN

Let S be a discrete semigroup and let the Stone–Čech compactification βS of S have the operation extending that of S which makes βS a right topological semigroup with S contained in its topological center. We show that the closure of the set of multiplicative idempotents in β N does not meet the set of additive idempotents in β N . We also show that the following algebraically defined subsets of β N are not Borel: the set of idempotents; the smallest ideal; any semiprincipal right ideal of N ⁎ ; the set of idempotents in any left ideal; and N ⁎ + N ⁎ . We extend these results to βS, where S is an infinite countable semigroup algebraically embeddable in a compact topological group

Recurrence in the dynamical system (X,〈Ts〉s∈S) and ideals of βS

A dynamical system is a pair ( X , 〈 T s 〉 s ∈ S ) , where X is a compact Hausdorff space, S is a semigroup, for each s ∈ S , T s is a continuous function from X to X , and for all s , t ∈ S , T s ∘ T t = T s t . Given a point p ∈ β S , the Stone-Čech compactification of the discrete space S , T p : X → X is defined by, for x ∈ X , T p ( x ) = p − lim s ∈ S T s ( x ) . We let β S have the operation extending the operation of S such that β S is a right topological semigroup and multiplication on the left by any point of S is continuous. Given p , q ∈ β S , T p ∘ T q = T p q , but T p is usually not continuous. Given a dynamical system ( X , 〈 T s 〉 s ∈ S ) , and a point x ∈ X , we let U ( x ) = p ∈ β S : T p ( x ) is uniformly recurrent . We show that each U ( x ) is a left ideal of β S and for any semigroup we can get a dynamical system with respect to which K ( β S ) = ⋂ x ∈ X U ( x ) and c ℓ K ( β S ) = ⋂ U ( x ) : x ∈ X and U ( x ) is closed . And we show that weak cancellation assumptions guarantee that each such U ( x ) properly contains K ( β S ) and has U ( x ) ∖ c ℓ K ( β S ) ≠ ∅

Dark Matter and the Chemical Evolution of Irregular Galaxies

We present three types of chemical evolution models for irregular galaxies: closed-box with continuous star formation rates (SFRs), closed-box with bursting SFRs, and O-rich outflow with continuous SFRs. We discuss the chemical evolution of the irregular galaxies NGC 1560 and II Zw 33, and a ``typical'' irregular galaxy. The fraction of low-mass stars needed by our models is larger than that derived for the solar vicinity, but similar to that found in globular clusters. For our typical irregular galaxy we need a mass fraction of about 40% in the form of substellar objects plus non baryonic dark matter inside the Holmberg radius, in good agreement with the results derived for NGC 1560 and II Zw 33 where we do have an independent estimate of the mass fraction in non baryonic dark matter. Closed-box models are better than O-rich outflow models in explaining the C/O and Z/O observed values for our typical irregular galaxy.Comment: 14 pages, 2 figure, uses emulateapj.sty package. ApJ in press. New models were added. The order of Tables has been correcte