Essentially tight bounds for rainbow cycles in proper edge-colourings

TL;DR

This work resolves, up to a $ ext{log} obreak ext{log} n$ factor, the Keevash–Mubayi–Sudakov–Verstraëte rainbow-cycle problem by proving an essentially tight upper bound of $O(( ext{log} obreak n)^{1+o(1)})$ on the maximum possible average degree of a properly edge-colored $n$-vertex graph with no rainbow cycle. The authors develop a robust sublinear expander framework and a novel two-coloring (palette-splitting) random process, together with a suite of probabilistic tools and auxiliary lemmas, to force the existence of a rainbow cycle under large average degree. The paper also reveals deep connections to additive number theory, showing that the rainbow-cycle bounds yield new bounds on additive dimension and plus-minus Davenport-type constants for both abelian and non-abelian groups, and it extends to transpositions in symmetric groups with near-optimal asymptotics. Overall, the results substantially advance our understanding of rainbow structures in edge-colored graphs and illuminate their additive-structure consequences. The findings have potential implications for rainbow Turán-type problems, expander methods in combinatorics, and group-theoretic dimension questions.

Abstract

An edge-coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge-coloured graph on $n$ vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of $(\log n)^{2+o(1)}$ for this question. Very recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the $o(1)$ term in Tomon's bound, showing a bound of $O(\log^2 n)$. We prove an upper bound of $(\log n)^{1+o(1)}$ for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the $o(1)$ term, and so it essentially resolves this question. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non-abelian groups.

Theorems & Definitions (40)

• Theorem 1.1
• Proposition 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Definition 3.1
• Lemma 3.2
• proof : Proof of \ref{['lemma-find-robust-expander-subgraph']}
• Lemma 4.1
• proof