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- W4288331328 abstract "We answer two extremal questions about odd cycles that naturally arise in the study of sparse pseudorandom graphs. Let $Gamma$ be an $(n,d,lambda)$-graph, i.e., $n$-vertex, $d$-regular graphs with all nontrivial eigenvalues in the interval $[-lambda,lambda]$. Krivelevich, Lee, and Sudakov conjectured that, whenever $lambda^{2k-1}ll d^{2k}/n$, every subgraph $G$ of $Gamma$ with $(1/2+o(1))e(Gamma)$ edges contains an odd cycle $C_{2k+1}$. Aigner-Horev, H`{a}n, and the third author proved a weaker statement by allowing an extra polylogarithmic factor in the assumption $lambda^{2k-1}ll d^{2k}/n$, but we completely remove it and hence settle the conjecture. This also generalises Sudakov, Szabo, and Vu's Tur'{a}n-type theorem for triangles. Secondly, we obtain a Ramsey multiplicity result for odd cycles. Namely, in the same range of parameters, we prove that every 2-edge-colouring of $Gamma$ contains at least $(1-o(1))2^{-2k}d^{2k+1}$ monochromatic copies of $C_{2k+1}$. Both results are asymptotically best possible by Alon and Kahale's construction of $C_{2k+1}$-free pseudorandom graphs." @default.
- W4288331328 created "2022-07-28" @default.
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- W4288331328 date "2019-06-12" @default.
- W4288331328 modified "2023-09-26" @default.
- W4288331328 title "Odd cycles in subgraphs of sparse pseudorandom graphs" @default.
- W4288331328 doi "https://doi.org/10.48550/arxiv.1906.05100" @default.
- W4288331328 hasPublicationYear "2019" @default.
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