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A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlos, Sarkozy and Szemeredi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph $H$ in a quasirandom host graph $G$, assuming that the edge-colouring of $G$ fulfills a boundedness condition that is asymptotically best possible. This has many applications beyond rainbow colourings, for example to graph decompositions, orthogonal double covers and graph labellings.
We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to longstanding decomposition problems. For instance, our results imply the following. Let $G$ be a quasi-random $n$-vertex graph and
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 qu
We study the mixed Ramsey number maxR(n,K_m,K_r), defined as the maximum number of colours in an edge-colouring of the complete graph K_n, such that K_n has no monochromatic complete subgraph on m vertices and no rainbow complete subgraph on r vertic
A subgraph $H$ of an edge-coloured graph is called rainbow if all of the edges of $H$ have different colours. In 1989, Andersen conjectured that every proper edge-colouring of $K_{n}$ admits a rainbow path of length $n-2$. We show that almost all opt
The blow-up lemma states that a system of super-regular pairs contains all bounded degree spanning graphs as subgraphs that embed into a corresponding system of complete pairs. This lemma has far-reaching applications in extremal combinatorics. We