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Off-diagonal Rado number for $x+y+c=z$ and $x+y+k=z$

Rajat Adak, Yash Bakshi, L. Sunil Chandran, Saraswati Girish Nanoti

Abstract

The study of Ramsey-type problems for linear equations originated with Schur's theorem and was later placed in a systematic framework by Richard Rado. In the off-diagonal setting, one fixes a pair of distinct linear equations $(\mathcal{E}_1, \mathcal{E}_2)$ and asks for the least integer $N$ such that every red--blue coloring of $\{1, 2, \dots, N\}$ must yield either a red solution to $\mathcal{E}_1$ or a blue solution to $\mathcal{E}_2$. This threshold integer is referred to as the off-diagonal Rado number of the system $(\mathcal{E}_1, \mathcal{E}_2)$. In this work, we study the discrete and continuous off-diagonal Rado number for non-homogeneous linear system of equations $x+y+c=z$ and $x+y+k=z$ where $c\le k$. We determine the exact two-color discrete and continuous off-diagonal Rado number $R_2(c,k)$ associated with this system of equations.

Off-diagonal Rado number for $x+y+c=z$ and $x+y+k=z$

Abstract

The study of Ramsey-type problems for linear equations originated with Schur's theorem and was later placed in a systematic framework by Richard Rado. In the off-diagonal setting, one fixes a pair of distinct linear equations and asks for the least integer such that every red--blue coloring of must yield either a red solution to or a blue solution to . This threshold integer is referred to as the off-diagonal Rado number of the system . In this work, we study the discrete and continuous off-diagonal Rado number for non-homogeneous linear system of equations and where . We determine the exact two-color discrete and continuous off-diagonal Rado number associated with this system of equations.

Paper Structure

This paper contains 6 sections, 2 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Main Result
  4. The discrete version
  5. The Continuous version
  6. Conclusion

Key Result

Theorem 1

Let $c$ and $k$ be positive integers with $1 \le c \le k$. Then the two-color off-diagonal Rado number $R_2(c,k)$ for the system is:

Theorems & Definitions (18)

  • Definition 1.1: Rado number
  • Definition 1.2: Off-diagonal Rado number
  • Definition 1.3: Off-diagonal continuous Rado number
  • Theorem 1
  • proof
  • Claim 3.1
  • proof
  • Claim 3.2
  • proof
  • Remark 3.3
  • ...and 8 more