Resolving a conjecture on degree of regularity of linear homogeneous equations
The electronic journal of combinatorics, Tome 21 (2014) no. 3
A linear equation is $r$-regular, if, for every $r$-coloring of the positive integers, there exist positive integers of the same color which satisfy the equation. In 2005, Fox and Radoićič conjectured that the equation $x_1 + 2x_2 + \cdots + 2^{n-2}x_{n-1} - 2^{n-1}x_n = 0$, for any $n \geq 2$, has a degree of regularity of $n-1$, which would verify a conjecture of Rado from 1933. Rado's conjecture has since been verified with a different family of equations. In this paper, we show that Fox and Radoićič's family of equations indeed have a degree of regularity of $n-1$. We also prove a few extensions of this result.
DOI :
10.37236/4321
Classification :
05D10, 05A17
Mots-clés : colorings, partition regularity, Ramsey theory
Mots-clés : colorings, partition regularity, Ramsey theory
@article{10_37236_4321,
author = {Noah Golowich},
title = {Resolving a conjecture on degree of regularity of linear homogeneous equations},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {3},
doi = {10.37236/4321},
zbl = {1298.05315},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4321/}
}
Noah Golowich. Resolving a conjecture on degree of regularity of linear homogeneous equations. The electronic journal of combinatorics, Tome 21 (2014) no. 3. doi: 10.37236/4321
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