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- W4304192755 abstract "An edge-colored graph $F$ is rainbow if each edge of $F$ has a unique color. The rainbow Tur'an number $ex^*(n,F)$ of a graph $F$ is the maximum possible number of edges in a properly edge-colored $n$-vertex graph with no rainbow copy of $F$. The study of rainbow Tur'an numbers was introduced by Keevash, Mubayi, Sudakov, and Verstraete in 2007. In this paper we focus on $ex^*(n,P_5)$. While several recent papers have investigated rainbow Tur'an numbers for $ell$-edge paths $P_{ell}$, exact results have only been obtained for $ell < 5$, and $P_5$ represents one of the smallest cases left open in rainbow Tur'{a}n theory. In this paper, we prove that $ex^*(n,P_5) leq frac{5n}{2}$. Combined with a lower-bound construction due to Johnston and Rombach, this result shows that $ex^*(n,P_5) = frac{5n}{2} $ when $n$ is divisible by $16$, thereby settling the question asymptotically for all $n$. In addition, this result strengthens the conjecture that $ex^*(n,P_{ell}) = frac{ell}{2}n + O(1)$ for all $ell geq 3$." @default.
- W4304192755 created "2022-10-11" @default.
- W4304192755 creator A5039196851 @default.
- W4304192755 date "2022-10-07" @default.
- W4304192755 modified "2023-09-27" @default.
- W4304192755 title "The rainbow Tur'an number of $P_5$" @default.
- W4304192755 doi "https://doi.org/10.48550/arxiv.2210.03376" @default.
- W4304192755 hasPublicationYear "2022" @default.
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