Ringel’s conjecture

Ringel posed the conjecture in $1963$: every tree with $n$ edges should decompose $K_{2n+1}$ into $2n+1$ copies. The asymptotic theorem is established, but the supplied evidence does not establish the remaining all-$n$ statement.

Known results

• Montgomery, Pokrovskiy, and Sudakov ($2020$; journal publication $2021$): for sufficiently large $n$, $K_{2n+1}$ decomposes into copies of every tree with $n$ edges.
• Keevash and Staden ($2020$): proved a stronger quasirandom-graph packing theorem, including the complete-graph case for sufficiently large $n$.
• Allen, Böttcher, Clemens, and Taraz ($2019$): obtained probabilistic packing results for many random trees, not the full conjecture.

August 2026 formalization

Walid Elkersh’s repository machine-checks graph-theoretic and probabilistic components of the sufficiently-large-$n$ proof development. This strengthens verification of the asymptotic result but does not settle the full all-$n$ conjecture; independent review of the repository is not documented in the supplied evidence.

Current status (as of August 2026): the sufficiently-large-$n$ case is proved and partly formalized, while the full all-$n$ conjecture remains unsettled.