A Short Proof of the Blow-Up Lemma for Approximate Decompositions

Kim, Kühn, Osthus and Tyomkyn (Trans. Amer. Math. Soc. 371 (2019), 4655–4742) greatly extended the well-known blow-up lemma of Komlós, Sárközy and Szemerédi by proving a ‘blow-up lemma for approximate decompositions’ which states that multipartite quasirandom graphs can be almost decomposed into any collection of bounded degree graphs with the same multipartite structure and slightly fewer edges. This result has already been used by Joos, Kim, Kühn and Osthus to prove the tree packing conjecture due to Gyárfás and Lehel from 1976 and Ringel’s conjecture from 1963 for bounded degree trees as well as implicitly in the recent resolution of the Oberwolfach problem (asked by Ringel in 1967) by Glock, Joos, Kim, Kühn and Osthus. Here we present a new and significantly shorter proof of the blow-up lemma for approximate decompositions. In fact, we prove a more general theorem that yields packings with stronger quasirandom properties which is useful for potential applications.