Ringel's conjecture (graceful decompositions)

For every positive integer $n$ and every tree $T$ with $n$ edges, the complete graph $K_{2n+1}$ can be decomposed into $2n+1$ edge-disjoint subgraphs each isomorphic to $T$.

Richard Montgomery, Alexey Pokrovskiy, and Benny Sudakov announced a proof in January 2020 (published in Geometric and Functional Analysis in 2021) establishing Ringel's conjecture for all sufficiently large $n$. Their proof finds a rainbow copy of any $n$-edge tree $T$ inside the natural-distance edge-coloring of $K_{2n+1}$—whose $2n+1$ cyclic shifts decompose $K_{2n+1}$—by combining deterministic embeddings of high-degree vertices, randomized embeddings preserving statistical independence, and distributive absorption. This resolved Ringel's conjecture asymptotically without settling the stronger Ringel–Kotzig graceful tree conjecture.