A rainbow blow-up lemma

TL;DR

This paper establishes a rainbow version of the blow-up lemma for bounded edge colourings, facilitating the study of rainbow embeddings and enabling applications like transferring the bandwidth theorem to the rainbow setting.

Contribution

It introduces a novel rainbow blow-up lemma for bounded edge colourings, extending the classical lemma to rainbow embeddings and related combinatorial applications.

Findings

Proves a rainbow blow-up lemma for . . -bounded edge colourings.

Enables systematic study of rainbow embeddings of bounded degree spanning subgraphs.

Applications include transferring the bandwidth theorem and rainbow decomposition results.

Abstract

We prove a rainbow version of the blow-up lemma of Koml\'os, S\'ark\"ozy and Szemer\'edi for $\mu n$-bounded edge colourings. This enables the systematic study of rainbow embeddings of bounded degree spanning subgraphs. As one application, we show how our blow-up lemma can be used to transfer the bandwidth theorem of B\"ottcher, Schacht and Taraz to the rainbow setting. It can also be employed as a tool beyond the setting of $\mu n$-bounded edge colourings. Kim, K\"uhn, Kupavskii and Osthus exploit this to prove several rainbow decomposition results. Our proof methods include the strategy of an alternative proof of the blow-up lemma given by R\"odl and Ruci\'nski, the switching method, and the partial resampling algorithm developed by Harris and Srinivasan.