# A rainbow blow-up lemma for almost optimally bounded edge-colourings

## TL;DR

This paper proves a rainbow blow-up lemma applicable to nearly optimally bounded edge-colourings, enabling the embedding of bounded-degree spanning subgraphs in quasirandom graphs with optimal colour-boundedness conditions.

## Contribution

It introduces a rainbow version of the blow-up lemma that works under near-optimal boundedness conditions, extending its applicability to various graph embedding problems.

## Key findings

- Existence of rainbow copies of bounded-degree spanning subgraphs in quasirandom graphs.
- The boundedness condition is asymptotically optimal.
- Applications to graph decompositions and labellings.

## Abstract

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 Koml\'os, S\'ark\"ozy and Szemer\'edi 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.