We show the existence of several infinite monochromatic patterns in the integers obtained as values of suitable symmetric polynomials; in particular, we obtain extensions of both the additive and multiplicative versions of Hindman's theorem. These configurations are obtained by means of suitable symmetric polynomials that mix the two operations. The simplest example is the following. For every finite coloring $\mathbb{N}=C_1\cup \dots \cup C_r$ there exists an infinite increasing sequence $a<b<c<\dots$ such that all elements below are monochromatic:$a,b,c,\dots ,a+b+ab,a+c+ac,b+c+bc,\dots ,a+b+c+ab+ac+bc+abc,\dots .$ The proofs use tools from algebra in the space of ultrafilters $\beta \mathbb{Z}$.

我们展示了作为合适对称多项式的值获得的整数中存在几个无限单色模式；特别是，我们获得了 Hindman 定理的加法和乘法版本的扩展。这些配置是通过混合这两种操作的合适的对称多项式来获得的。最简单的例子如下。对于每一个有限的着色$\mathbb{ñ}=C_1\cup \dots \cup C_r$存在一个无限递增的序列$\mathrm{一种}<b<C<\dots$使得以下所有元素都是单色的：$\mathrm{一种},b,C,\dots ,\mathrm{一种}+b+\mathrm{一种}b,\mathrm{一种}+C+\mathrm{一种}C,b+C+bC,\dots ,\mathrm{一种}+b+C+\mathrm{一种}b+\mathrm{一种}C+bC+\mathrm{一种}bC,\dots .$证明使用超滤器空间中的代数工具$\beta \mathbb{Z}$.