Hilbert versus Hindman
It is well known that Hindman's theorem can (equivalently) be formulated in the following finite union form: Let \(f: \mathbb{N^{<N}} \to k\) be a finite coloring of the finite subsets of \(\mathbb{N}\). Then there exists an infinite sequence of finite subsets \((X_i)_{i\in\mathbb{N}}\) such that the sets \(X_i\) are unmeshed, i.e., for all \(i,j\) either \(\max(X_i) < \min(X_{j})\) or \(\max(X_j) < \min(X_i)\), and that there is a \(c\) such that for all finite sets \(F\) we have \(f(\bigcup_{i\in F} X_i) = c\). In this paper the author considers the variant of this theorem where `unmeshed' is relaxed to `distinct'. He shows that this variant is equivalent to the infinite pigeonhole principle and thus to the \(\Pi^0_1\)-bounded collection principle (\(\mathrm{B}\Pi^0_1\)). This shows that this variant of Hindman's theorem is finitistic in the sense of Hilbert, while Hindman's theorem is not.
scientific article; zbMATH DE number 6005023