= Partition regularity of IP sets
Every <finite coloring> of an <IP set> has a color class that is an <IP set>. Here is a reduction to the ordinary <Hindman theorem>. Given $\operatorname{FS}(x_i)$ in the original set, color an integer $n>0$ by the color of $\sum_{i\in\operatorname{supp}_2(n)}x_i$, where the support consists of positions of its nonzero digits in the <binary expansion>. The <Hindman theorem> produces $w_i>0$ with <monochromatic> $\operatorname{FS}(w_i)$. Replace the $w_i$ by consecutive, disjoint finite block sums $v_j$ with separated binary supports: after a block has been chosen, take $K$ above all previously used positions; two of $2^K+1$ partial sums of a fresh tail agree modulo $2^K$, so their positive difference supplies the next block sum divisible by $2^K$. Now binary addition of distinct $v_j$ has no carries. Therefore $y_j=\sum_{i\in\operatorname{supp}_2(v_j)}x_i$ generates a <monochromatic> <finite-sums set> in the original IP set.
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