Monochromatic homothetic copies\\ of $\{1,1+s,1+s+t\}$

http://dx.doi.org/10.4153/CMB-1997-018-3
Canad. Math. Bull. 40(1997), 149-157

  Published:1997-06-01
 Printed: Jun 1997

For positive integers $s$ and $t$, let $f(s, t)$ denote the smallest positive integer $N$ such that every $2$-colouring of $[1,N]=\{1,2, \ldots , N\}$ has a monochromatic homothetic copy of $\{1, 1+s, 1+s+t\}$. We show that $f(s, t) = 4(s+t) + 1$ whenever $s/g$ and $t/g$ are not congruent to $0$ (modulo $4$), where $g=\gcd(s,t)$. This can be viewed as a generalization of part of van~der~Waerden's theorem on arithmetic progressions, since the $3$-term arithmetic progressions are the homothetic copies of $\{1, 1+1, 1+1+1\}$. We also show that $f(s, t) = 4(s+t) + 1$ in many other cases (for example, whenever $s > 2t > 2$ and $t$ does not divide $s$), and that $f(s, t) \le 4(s+t) + 1$ for all $s$, $t$. Thus the set of homothetic copies of $\{1, 1+s, 1+s+t\}$ is a set of triples with a particularly simple Ramsey function (at least for the case of two colours), and one wonders what other ``natural'' sets of triples, quadruples, {\it etc.}, have simple (or easily estimated) Ramsey functions.