Improved Two-Colour Rado Numbers for Linear Equations with Certain Coefficients
The electronic journal of combinatorics, Tome 32 (2025) no. 4
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.
@article{10_37236_13596,
author = {Ishan Arora and Srashti Dwivedi and Amitabha Tripathi},
title = {Improved {Two-Colour} {Rado} {Numbers} for {Linear} {Equations} with {Certain} {Coefficients}},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {4},
doi = {10.37236/13596},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13596/}
}
TY - JOUR AU - Ishan Arora AU - Srashti Dwivedi AU - Amitabha Tripathi TI - Improved Two-Colour Rado Numbers for Linear Equations with Certain Coefficients JO - The electronic journal of combinatorics PY - 2025 VL - 32 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.37236/13596/ DO - 10.37236/13596 ID - 10_37236_13596 ER -
%0 Journal Article %A Ishan Arora %A Srashti Dwivedi %A Amitabha Tripathi %T Improved Two-Colour Rado Numbers for Linear Equations with Certain Coefficients %J The electronic journal of combinatorics %D 2025 %V 32 %N 4 %U http://geodesic.mathdoc.fr/articles/10.37236/13596/ %R 10.37236/13596 %F 10_37236_13596
Ishan Arora; Srashti Dwivedi; Amitabha Tripathi. Improved Two-Colour Rado Numbers for Linear Equations with Certain Coefficients. The electronic journal of combinatorics, Tome 32 (2025) no. 4. doi: 10.37236/13596
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