Gold Partition Conjecture

Peczarski introduced the Gold Partition Conjecture in 2006 as a strengthening of the $1/3$–$2/3$ conjecture for finite posets. The general conjecture remains unresolved.

Known results

• Peczarski, 2006: proved the conjecture for posets with at most $11$ elements and for $6$-thin posets.
• Peczarski, 2006: exhaustive verification was later extended through $1111$ elements.
• Dolores-Cuenca, Guzmán-Sáenz, and Kim, 2024: proved preservation under lexicographic substitution and derived consequences for the $1/3$–$2/3$ conjecture.

July–September 2026 developments

In July 2026, Anish Gupta claimed a computer-assisted verification for every non-chain poset with at most $1414$ elements, including certificates for all $1{,}338{,}193{,}159{,}771$ unlabeled posets of order $14$; the computation is not independently verified here. On September 29, 2026, Order published The Gold Partition Conjecture and the Lexicographic Sum of Posets, but the retrieved record gives no abstract or assessable result.

Current status (as of September 2026): finite cases are claimed through $1414$ elements, and structural closure is known, but the general Gold Partition Conjecture remains open; the computational claim is unverified.