## Monochromatic forests of finite subsets of $$\mathbb{N}$$.(English)Zbl 0956.05098

The paper generalizes a known result on the existence of monochromatic, piecewise syndetic subsets of finite colorings of $$\mathbb{N}$$ to finite colorings of the set of all finite subsets of $$\mathbb{N}$$. It is shown that for any finite coloring of the set of all finite subsets of $$\mathbb{N}$$ and for any finite forest $$F$$, there exists a monochromatic $$d$$-copy of $$F$$. Several generalizations and modifications of this result are obtained by using Ramsey’s theorem and van der Waerden’s theorem, among others. The paper is concluded with suggestions for possible applications and open questions.

### MSC:

 05D10 Ramsey theory

### Keywords:

Ramsey theory; piecewise syndetic; algorithmic progression
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