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- W4288408896 abstract "Given a graph $H$, the extremal number $mathrm{ex}(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing $K'_{s,t}$ for the subdivision of the bipartite graph $K_{s,t}$, we show that $mathrm{ex}(n, K'_{s,t}) = O(n^{3/2 - frac{1}{2s}})$. This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for $t$ sufficiently large in terms of $s$. Second, for any integers $s, k geq 1$, we show that $mathrm{ex}(n, L) = Theta(n^{1 + frac{s}{sk+1}})$ for a particular graph $L$ depending on $s$ and $k$, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of ErdH{o}s and Simonovits, which asserts that every rational number $r in (1,2)$ is realisable in the sense that $mathrm{ex}(n,H) = Theta(n^r)$ for some appropriate graph $H$, giving infinitely many new realisable exponents and implying that $1 + 1/k$ is a limit point of realisable exponents for all $k geq 1$. Writing $H^k$ for the $k$-subdivision of a graph $H$, this result also implies that for any bipartite graph $H$ and any $k$, there exists $delta > 0$ such that $mathrm{ex}(n,H^{k-1}) = O(n^{1 + 1/k - delta})$, partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph $H$ with maximum degree $r$ on one side which does not contain $C_4$ as a subgraph satisfies $mathrm{ex}(n, H) = o(n^{2 - 1/r})$." @default.
- W4288408896 created "2022-07-29" @default.
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- W4288408896 date "2019-03-25" @default.
- W4288408896 modified "2023-09-30" @default.
- W4288408896 title "More on the extremal number of subdivisions" @default.
- W4288408896 doi "https://doi.org/10.48550/arxiv.1903.10631" @default.
- W4288408896 hasPublicationYear "2019" @default.
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