2) V2 For every finite chain of finite sets and every two-coloring, one color contains a subfamily with at least half the chain; that subfamily is automatically union-closed.
kernel-checked, filed Tue Aug 25 2026 07:22:55 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Scope. All finite chains of Finsets over arbitrary decidable ground types; the conclusion records containment, monochromaticity, binary union closure, and the exact factor-two cardinal bound.
1) V1 For every fixed natural exponent d, all sufficiently large n have the following property: every two-coloring of the subsets of [n] contains a monochromatic union-closed family with more than n^d members.
open, filed Tue Aug 25 2026 07:22:24 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Full local mode. Writer used the 1978 source and current problem page; the 2026 paper was opened before narrowing the scope. Independent transcription bridges both ways, constant colorings witness the domain, the exact negation is a fixed polynomial obstruction on arbitrarily large dimensions, and all eleven degenerates red as restatements. The whole attack kernel-checks closure of all nonempty unions generated by fixed atoms and identifies the quantitative bottleneck: fixed-k finite-unions Ramsey results do not provide k growing faster than each constant times log n. A separate chain-pigeonhole theorem preflights green and recovers the classical linear route. Vendor diversity was unavailable; sources varied by role.
Scope. The surviving superpolynomial lower-bound conjecture for F(n); subsets of [n] are Finset (Fin n), colors are Bool, closure includes the union of every two members.