It's not always true . Assume $n=b.3^a$ where $(3,b)=1$, We can color $n$ with one of the $4$ colors due to reminder $b$ when divided by $3$ and parity of $a$ . ($4=2 \times 2$) This argument works for every prime $p \geq 3$ ($x+y=pz$) , but it fails at $p=2$ ......Why ? Because it's always true that for every $r-$coloring of positive integers , there is a monochromatic solution to $x+y=2z$ , It's a consequence of celebrated Van der Waerden's theorem .

The equation writes $x + y -pz = 0$, for $p\geq 3$. Rado calls regular a linear equation $\sum a_ix_i$, where the coefficients $a_i$ are non-zero integers, if for any $r$-coloring of the natural numbers the equation has a monochromatic solution. He has proved that an equation is regular if and only if $\sum a_j$ for some subset of the coefficients. Since this is not true for $\{1,1,-p\}$, it means our equation is not regular. The fact it is regular for $p=2$ can now be seen as Rado's generalization of Van der Waerden's theorem. (Thus, in Mongolia at least, Rado's results are folklore ...).

mavropnevma wrote: The equation writes $x + y -pz = 0$, for $p\geq 3$. Rado calls regular a linear equation $\sum a_ix_i$, where the coefficients $a_i$ are non-zero integers, if for any $r$-coloring of the natural numbers the equation has a monochromatic solution. He has proved that an equation is regular if and only if $\sum a_j$ for some subset of the coefficients. Since this is not true for $\{1,1,-p\}$, it means our equation is not regular. The fact it is regular for $p=2$ can now be seen as Rado's generalization of Van der Waerden's theorem. (Thus, in Mongolia at least, Rado's results are folklore ...). yes,it is connected with Rado's theorem. when $k=1$,it's Schur's theorem;when $k=2$,it's a weak conjecture of Vanderwarden's theorem. but why do you say it's 'folklore'?

Bacteria wrote: (proposed by B. Batbaysgalan, folklore) That is how Mr. Batbaysgalan considered it ...