Canonical Theorems for Colored Integers with Respect to Some Linear Combinations

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-02-01 DOI:10.1137/21m1454195

Maria Axenovich, Hanno Lefmann

引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 609-628, March 2024.
Abstract. Hindman proved in 1979 that no matter how natural numbers are colored in [math] colors, for a fixed positive integer [math], there is an infinite subset [math] of numbers and a color [math] such that for any finite nonempty subset [math] of [math], the color of the sum of elements from [math] is [math]. Later, Taylor extended this result to colorings with an unrestricted number of colors and five unavoidable color patterns on finite sums. This result is referred to as a canonization of Hindman’s theorem and parallels the canonical Ramsey theorem of Erdős and Rado. We extend Taylor’s result from sums, that are linear combinations with coefficients 1, to several linear combinations with coefficients 1 and [math]. These results in turn could be interpreted as canonical-type theorems for solutions to infinite systems.

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关于某些线性组合的有色整数典范定理

SIAM 离散数学杂志》，第 38 卷第 1 期，第 609-628 页，2024 年 3 月。摘要。欣德曼在 1979 年证明，无论自然数如何用 [math] 颜色着色，对于一个固定的正整数 [math]，存在一个无限的数子集 [math] 和一种颜色 [math]，使得对于 [math] 的任何有限非空子集 [math]，来自 [math] 的元素之和的颜色是 [math]。后来，泰勒将这一结果扩展到具有不受限制的颜色数和有限和上五个不可避免的颜色模式的着色。这一结果被称为辛德曼定理的典范化，与厄多斯和拉多的典范拉姆齐定理相似。我们将泰勒的结果从系数为 1 的线性组合和扩展到系数为 1 和 [math] 的多个线性组合。这些结果反过来又可以解释为无限系统解的典型定理。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.