Generalized rainbow Tur\'an numbers of odd cycles

TL;DR

This paper establishes upper bounds on the maximum number of triangles in properly edge-colored graphs avoiding rainbow odd cycles, extending previous results on rainbow cycles and matching known lower bounds.

Contribution

It proves the upper bound for rainbow triangle counts in graphs avoiding rainbow odd cycles, filling a gap in the understanding of rainbow Turán numbers for these structures.

Findings

Upper bound for $ ext{ex}(n,C_3, ext{rainbow-}C_{2k+1})$ established as $O(n^{1+1/k})

Matches known lower bounds for even $k$, conjectured to be tight for odd $k$

Extends previous work on rainbow cycle Turán numbers to odd cycles

Abstract

Given graphs $F$ and $H$, the generalized rainbow Tur\'an number $\text{ex}(n,F,\text{rainbow-}H)$ is the maximum number of copies of $F$ in an $n$-vertex graph with a proper edge-coloring that contains no rainbow copy of $H$. B. Janzer determined the order of magnitude of $\text{ex}(n,C_s,\text{rainbow-}C_t)$ for all $s\geq 4$ and $t\geq 3$, and a recent result of O. Janzer implied that $\text{ex}(n,C_3,\text{rainbow-}C_{2k})=O(n^{1+1/k})$. We prove the corresponding upper bound for the remaining cases, showing that $\text{ex}(n,C_3,\text{rainbow-}C_{2k+1})=O(n^{1+1/k})$. This matches the known lower bound for $k$ even and is conjectured to be tight for $k$ odd.