A 2-coloring of $[1,n]$ can have $\frac{n^2}{2a(a^2+2a+3)} + O(n)$ monochromatic triples of the form $\{x,y,x+ay\}, a \geq 2$, but not less!

Details A 2-coloring of $[1,n]$ can have $\frac{n^2}{2a(a^2+2a+3)} + O(n)$ monochromatic triples of the form $\{x,y,x+ay\}, a \geq 2$, but not less! A 2-coloring of $[1,n]$ can have $\frac{n^2}{2a(a^2+2a+3)} + O(n)$ monochromatic triples of the form $\{x,y,x+ay\}, a \geq 2$, but not less!

2008-01-05

arxiv.org/abs/0801.0798v1

Mathematics

Combinatorics

10 pages, 3 fugures

Scientific paper

We solve a problem posed by Ronald Graham about the minimum number, over all
2-colorings of $[1,n]$, of monochromatic $(x,y,x+ay)$ triples, $a \geq 2$. We
show that the minimum number of such triples is $\frac{n^2}{2a(a^2+2a+3)} +
O(n)$. We also find a new upper bound for the minimum number, over all
$r$-colorings of $[1,n]$, of monochromatic Schur triples, for $r \geq 3$.

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