On the strength of Hindman's Theorem for bounded sums or unions

We report on recent results on the strength of restrictions of Hindman's Theorem to finite sums or unions, and variants theoreof. On the one hand we show that some weak restrictions already attain the known ACA$_0$ lower bound for the full theorem. On the other hand we show that some other weak restrictions also admit an ACA$_0$ upper bound, which is much lower than the only upper bound known on the full theorem, i.e. ACA$_0^+$. We also discuss connections with the Increasing Polarized Ramsey's Theorem and highlight the role of a sparsity condition on the solution set, which we call apartness. Part of this work is join with Kolodziejczyk, Lepore and Zdanowski.