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

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 Turán problems, a framework systematically introduced by Keevash, Mubayi, Sudakov and Verstraëte [\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, Bucić, 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.

💡 Research Summary

The paper investigates a rainbow analogue of Mader’s classic theorem on subdivisions in dense graphs. Mader proved that any graph on (n) vertices with average degree (\Omega(t^{2})) contains a subdivision of the complete graph (K_{t}). The authors ask the same question for properly edge‑coloured graphs, where each vertex sees incident edges of distinct colours, and a subgraph is called rainbow if all its edges have different colours.

The main result (Theorem 1.4) states that if a properly edge‑coloured graph (G) on (n) vertices satisfies
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