论文标题
广义的Rado和Roth标准
Generalised Rado and Roth criteria
论文作者
论文摘要
我们研究方程式的ramsey属性$ a_1p（x_1） + \ cdots + a_sp（x_s）= b $，其中$ a_1，\ ldots，a_s，b $是整数，$ p $是$ d $ $ d $的Integer polledenomial golynomial fornemial。只要至少有$（1+o（1））d^2 $变量，我们表明Rado的标准和相交条件完全表征了该形式的方程哪些方程允许单色解决方案相对于正整数的任意有限颜色。此外，我们获得了这些方程式的Roth型定理，这表明它们在且仅当$ B = a_1 + \ cdots + a_s = 0 $时，在任何具有正密度的整体上都接受了非恒定解决方案。此外，我们为单色/致密溶液的数量（过饱和）建立了尖锐的渐近下限。
We study the Ramsey properties of equations $a_1P(x_1) + \cdots + a_sP(x_s) = b$, where $a_1,\ldots,a_s,b$ are integers, and $P$ is an integer polynomial of degree $d$. Provided there are at least $(1+o(1))d^2$ variables, we show that Rado's criterion and an intersectivity condition completely characterise which equations of this form admit monochromatic solutions with respect to an arbitrary finite colouring of the positive integers. Furthermore, we obtain a Roth-type theorem for these equations, showing that they admit non-constant solutions over any set of integers with positive upper density if and only if $b= a_1 + \cdots + a_s = 0$. In addition, we establish sharp asymptotic lower bounds for the number of monochromatic/dense solutions (supersaturation).
