Monochromatic solutions to equations with unit fractions

Source abstract

Our main result is that if G ( x 1 , …, x n ) = 0 is a system of homogeneous equations such that for every partition of the positive integers into finitely many classes there are distinct y 1 ,…, y n in one class such that G ( y 1 , …, y n ) = 0, then, for every partition of the positive integers into finitely many classes there are distinct Z 1 , …, Z n in one class such that In particular, we show that if the positive integers are split into r classes, then for every n ≥ 2 there are distinct positive integers x 1 , x 1 , …, x n in one class such that We also show that if [1, n 6 − ( n 2 − n ) 2 ] is partitioned into two classes, then some class contains x 0 , x 1 , …, x n such that (Here, x 0 , x 2 , …, x n are not necessarily distinct.)