Nearly tight bound for rainbow clique subdivisions in properly edge-colored graphs and applications

TL;DR

This paper establishes a nearly tight bound for the existence of rainbow clique subdivisions in properly edge-colored graphs with large average degree, resolving a key open question and connecting to various fields.

Contribution

It proves a tight bound for rainbow clique subdivisions in properly edge-colored graphs, advancing understanding of rainbow analogues of classical graph theorems.

Findings

Every properly edge-colored graph with average degree at least t^2(log n)^{1+o(1)} contains a rainbow subdivision of K_t.

The bound is tight up to lower order terms for fixed t.

Applications to additive combinatorics, number theory, and coding theory are demonstrated.

Abstract

An edge-colored graph is said to be rainbow if all its edges have distinct colors. In this paper, we study the rainbow analogue of a fundamental result of Mader [\emph{Math. Ann.} \textbf{174} (1967), 265--268] on the existence of subdivisions in graphs with large average degree. This is part of the study of rainbow analogues of classical Tur\'an problems, a framework systematically introduced by Keevash, Mubayi, Sudakov and Verstra\"ete [\emph{Combin. Probab. Comput.} \textbf{16} (2007), 109--126]. We prove that every properly edge-colored graph on $n$ vertices with average degree at least $t^2(\log n)^{1+o(1)}$ contains a rainbow subdivision of $K_t$. When $t$ is a constant, this bound is tight up to the $o(1)$ term. So it essentially resolves a question raised by Jiang, Methuku and Yepremyan [\emph{European J. Combin.} \textbf{110} (2023), 103675] on rainbow clique subdivisions, and also implies a result of Alon, Buci\'c, Sauermann, Zakharov and Zamir [\emph{Proc. Lond. Math. Soc.} \textbf{130} (2025), e70044] on rainbow cycles. In addition, we present several applications of our result to problems in additive combinatorics, number theory and coding theory.