A proof of Ringel’s conjecture

Abstract

A typical decomposition question asks whether the edges of some graph G can be partitioned into disjoint copies of another graph H. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with n edges packs $$2n+1$$ 2 n + 1 times into the complete graph $$K_{2n+1}$$ K 2 n + 1 . In this paper, we prove this conjecture for large n.

Transference for loose Hamilton cycles in random $3$-uniform hypergraphs

“…The absorbing method was initially explicitly introduced by Rödl, Ruciński, and Szemerédi [61] even though the implicit idea has its roots in the works of Krivelevich [44] and Erdős, Gyárfás, and Pyber [16]. Recently it has seen a surge of interest and has been used in a variety of settings: combinatorial designs [27,35], decompositions [26,48], Steiner systems [50,19], Ramsey theory [9,37], colouring (hyper)graphs [54,33], embeddings [53,24], and many, many more.…”

The Unity of Combinatorics

“…Same with the Ringel conjecture on packing complete graphs 𝐾 2𝑛+1 with (2𝑛 + 1) isomorphic trees on 𝑛 vertices, recently resolved for large 𝑛 in a fantastic development by Richard Montgomery, Alexey Pokrovskiy, and Benny Sudakov [8]. By comparison, the book opted for a nice observation on how to use Steiner triple systems to pack 100 triangles into 𝐾 25 (see §6.5).…”

On f-derangements and decomposing bipartite graphs into paths

“…Ringel's Conjecture was recently proved for sufficiently large graphs [6] but both these Conjectures appear to be far from fully proved. There have been hundreds or even thousands of papers devoted to them.…”