Abstract:We prove a theorem ensuring that the compositions of certain Ramsey families are still Ramsey. As an application, we show that in any finite coloring of $\mathbb{N}$ there is an infinite set $A$ and an as large as desired finite set $B$ with $(A+B)\cup (AB)$ monochromatic, addressing a problem of Kra, Moreira, Richter, and Robertson. In fact, we prove an iterated version of this result, which ensures the existence of monochromatic patterns such as $\{a\circ_1 (b \circ_2 c): \circ_i\in \{+,\cdot\}\}, $ generalizing a Ramsey theorem of Bergelson and Moreira that was previously only known to hold for colorings of $\mathbb{Q}$ rather than colorings of $\mathbb{N}$. Our main new technique is an extension of the color focusing method that involves trees rather than sequences.
From: Matthew Bowen [view email]
[v1]
Tue, 25 Oct 2022 20:13:08 UTC (137 KB)
[v2]
Mon, 21 Nov 2022 15:13:12 UTC (137 KB)
[v3]
Sat, 25 Jul 2026 17:27:32 UTC (146 KB)
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