Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the two-colour Rado number for $\sum_{i=1}^m a_ix_i=c$

Ishan Arora, Srashti Dwivedi, Amitabha Tripathi

Abstract

Let $a_1,\ldots,a_m$ be nonzero integers, $c \in \mathbb Z$ and $r \ge 2$. The Rado number for the equation \[ \sum_{i=1}^m a_ix_i = c \] in $r$ colours is the least positive integer $N$ such that any $r$-colouring of the integers in the interval $[1,N]$ admits a monochromatic solution to the given equation. We introduce the concept of $t$-distributability of sets of positive integers, and determine exact values whenever possible, and upper and lower bounds otherwise, for the Rado numbers when the set $\{a_1,\ldots,a_{m-1}\}$ is $2$-distributable or $3$-distributable, $a_m=-1$, and $r=2$. This generalizes previous works by several authors.

On the two-colour Rado number for $\sum_{i=1}^m a_ix_i=c$

Abstract

Let be nonzero integers, and . The Rado number for the equation in colours is the least positive integer such that any -colouring of the integers in the interval admits a monochromatic solution to the given equation. We introduce the concept of -distributability of sets of positive integers, and determine exact values whenever possible, and upper and lower bounds otherwise, for the Rado numbers when the set is -distributable or -distributable, , and . This generalizes previous works by several authors.

Paper Structure

This paper contains 3 sections, 11 theorems, 27 equations, 1 table.

Table of Contents

  1. Introduction
  2. Distributable Sets
  3. Main Results

Key Result

Proposition 2

(Brown, 1961) A sequence $\{u_n\}_{n \ge 1}$ of positive integers is complete if and only if where $s_k=u_1+\cdots+u_k$ for $k \ge 1$.

Theorems & Definitions (16)

  • Definition 1
  • Proposition 2
  • Definition 3
  • Definition 4
  • Remark 5
  • Proposition 6
  • Remark 7
  • Theorem 8
  • Corollary 9
  • Theorem 10
  • ...and 6 more