Worked solution: A rainbow-tree proof of Ringel's conjecture (2020)

Montgomery, Pokrovskiy and Sudakov (2021, Introduction) recall that Ringel posed Conjecture 1.1 in 1963: $K_{2n+1}$ can be decomposed into copies of any tree with $n$ edges. This is one of the oldest and best-known open problems on graph decompositions, verified previously only for special tree shapes (caterpillars, trees with at most $4$ leaves, trees with at most $35$ vertices, and others) or with an approximate/bounded-degree restriction (Joos-Kim-Kuhn-Osthus proved it for bounded-degree trees; Ferber-Samotij and Adamaszek-Allen-Grosu-Hladky proved approximate versions for maximum degree $O(n/\log n)$). This paper's main theorem removes every restriction on the tree's shape or degree: for every sufficiently large $n$, $K_{2n+1}$ decomposes into $2n+1$ copies of any tree $T$ with $n$ edges (Theorem 1.2).