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

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

Abstract

Ramsey-type problems for linear equations began with Schur's theorem and were systematically generalized by Richard Rado. In the off-diagonal framework for two colors, one considers two different linear equations $(\mathcal{E}_1,\mathcal{E}_2)$ and determines the minimum integer $N$ for which any red-blue coloring of $\{1,2,...,N\}$ forces either a red solution of the equation $\mathcal{E}_1$ or a blue solution of the equation $\mathcal{E}_2$. In this work, we study off-diagonal Rado numbers for non-homogeneous linear equations of the forms $x+y+c=z$ and $x+qy=z$. We determine the exact two-color off-diagonal Rado number $R_2(c,q)$ associated with this system of equations.

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

Abstract

Ramsey-type problems for linear equations began with Schur's theorem and were systematically generalized by Richard Rado. In the off-diagonal framework for two colors, one considers two different linear equations and determines the minimum integer for which any red-blue coloring of forces either a red solution of the equation or a blue solution of the equation . In this work, we study off-diagonal Rado numbers for non-homogeneous linear equations of the forms and . We determine the exact two-color off-diagonal Rado number associated with this system of equations.
Paper Structure (4 sections, 1 theorem, 19 equations)

This paper contains 4 sections, 1 theorem, 19 equations.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Main Result
  4. Conclusion

Key Result

Theorem 1

Let $c\ge1$ and $q\ge 1$ be positive integers. Then the two-color off-diagonal Rado number $R_2(c,q)$ for the system is:

Theorems & Definitions (14)

  • Definition 1.1: Rado number
  • Definition 1.2: Off-diagonal Rado number
  • Theorem 1
  • proof
  • Claim 3.1
  • proof
  • Claim 3.2
  • proof
  • Remark 3.3
  • Claim 3.4
  • ...and 4 more