A short proof of the blow-up lemma for approximate decompositions

Kim, Kuhn, Osthus and Tyomkyn (Trans. Amer. Math. Soc. 371 (2019), 4655--4742) greatly extended the well-known blow-up lemma of Komlos, Sarkozy and Szemeredi 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, Kuhn and Osthus to prove the tree packing conjecture due to Gyarfas and Lehel from 1976 and Ringels 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, Kuhn 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 so that it can be combined with Keevashs results on designs to obtain results of the following form. For all $varepsilon>0$, $rin mathbb{N}$ and all large $n$ (such that $r$ divides $n-1$), there is a decomposition of $K_n$ into any collection of $r$-regular graphs $H_1,ldots,H_{(n-1)/r}$ on $n$ vertices provided that $H_1,ldots,H_{varepsilon n}$ contain each at least $varepsilon n$ vertices in components of size at most $varepsilon^{-1}$.