Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2306.10895

On the construction of a family of sets of positive integers closed under taking subsets

Shoichi Kamada
View original source ↗Read source document ↗Export metadata ↓

Abstract

In the several contexts such as combinatorial number theory, families of sets of positive integers closed under taking subsets have been investigated. Then it is sometimes useful to give bijections between the set of the one-sided infinite sequences on the alphabet set $\{0,1\}$ and such a family of sets. The most typical example is the family of sum-free sets. Although such a kind of families covers a large class of families of sets, there are only a few considerations on bijections for the case where the sum-free property is replaced by another property. In this paper, we explicitly give a bijection and its inverse between the set of one-sided infinite sequences on the alphabet set $\{0,1\}$ and a family of sets which may be contained in a class of families closed under taking subsets. Moreover, we show that some extremal property in a particular family of sets is characterized by a discrete dynamical system based on this kind of bijections.

Explore related subjects

math.COmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shoichi Kamada. 2024-12-30. On the construction of a family of sets of positive integers closed under taking subsets. https://arxiv.org/abs/2306.10895

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Small doublings in abelian groups of prime power torsion

Let $A$ be a subset of $G$, where $G$ is a finite abelian group of torsion $r$. It was conjectured by Ruzsa that if $|A+A|\leq K|A|$, then $A$ is contained in a coset of $G$ of size at most $r^{CK}|A|$ for some constant $C$. The case $r=2$ received considerable attention in a sequence of papers, and was resolved by Green and Tao. Recently, Even-Zohar and Lovett settled the case when $r$ is a prime. In this paper, we confirm the conjecture when $r$ is a power of prime. In particular, the bound we obtain is tight.

math.CO↗
arXiv2026

Graph Polynomial for Colored Embedded Graphs: A Topological Approach

We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in physics. We also analyze a variant of these polynomials for colored embedded graphs. This is used to describe the change in the polynomial under basic graph theoretic operations. We conclude with several applications of this polynomial including detection of certain classes of graphs and the connection of this polynomial with topological entanglement entropy.

math.CO↗
arXiv2026

Schur positivity of the spiders $S(a,2,1)$ and $S(a,4,1)$ via noncommutative symmetric functions

We prove that the spider graphs $S(a,2,1)$ and $S(a,4,1)$ are Schur positive for all integers $a\ge1$. Together with the known $e$-positivity results, this completes the $e$- and Schur-positivity classification of both families. Our approach uses noncommutative symmetric functions, including a particularly simple ribbon expansion for the path lift with coefficients given by powers of two. We give a new proof of the Shareshian--Wachs path formula at $t=1$ and construct corresponding lifts for spiders. The Littlewood--Richardson rule converts their ribbon expansions into a general Schur-coefficient formula in terms of weighted Yamanouchi words. Mass-preserving multi-injections and a reduction to finitely many inequalities in degree $10$ then prove the required positivity; exact computer verification of these inequalities completes the proof in full generality.

math.CO↗