论文标题
关于理由中的单色总和和产品的一些新结果
Some New Results on Monochromatic Sums and Products in the Rationals
论文作者
论文摘要
我们在本文中的目的是表明，对于任何$ k $，都有有限的颜色，其分母仅包含第一个$ k $ primes，因此没有无限套件具有其所有有限的总和和产品单色。实际上，我们证明了这种“统一”形式：理性的有限色，没有无限集的属性，其分母仅包含有限的许多素数具有其所有有限的总和和产品单色。我们还提供了其他各种结果，包括新的简短证明了旧结果，即自然材料有有限的颜色，以至于没有无限套件具有其所有成对总和和单色的产品。
Our aim in this paper is to show that, for any $k$, there is a finite colouring of the set of rationals whose denominators contain only the first $k$ primes such that no infinite set has all of its finite sums and products monochromatic. We actually prove a `uniform' form of this: there is a finite colouring of the rationals with the property that no infinite set whose denominators contain only finitely many primes has all of its finite sums and products monochromatic. We also give various other results, including a new short proof of the old result that there is a finite colouring of the naturals such that no infinite set has all of its pairwise sums and products monochromatic.
