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# Journal of Combinatorics

## Volume 5 (2014)

### Number 2

### Degree of regularity of linear homogeneous equations and inequalities

Pages: 235 – 243

DOI: http://dx.doi.org/10.4310/JOC.2014.v5.n2.a5

#### Authors

#### Abstract

We define a linear homogeneous equation to be *strongly* $\mathrm{r}$-*regular* if, when a finite number of inequalities is added to the equation, the system of the equation and inequalities is still $r$-regular. In this paper, we show that if a linear homogeneous equation is $r$-regular, then it is strongly $r$-regular. In 2009, Alexeev and Tsimerman introduced a family of equations, each of which is $(n - 1)$-regular but not $n$-regular, verifying a conjecture of Rado from 1933. These equations are actually strongly $(n - 1)$-regular as an immediate corollary of our results.

#### Keywords

colorings, partition regularity, Ramsey theory

#### 2010 Mathematics Subject Classification

05D10