• Corpus ID: 119326112

# On the rational Tur\'an exponents conjecture

@article{Kang2018OnTR,
title={On the rational Tur\'an exponents conjecture},
author={Dong Yeap Kang and Jaehoon Kim and Hong Liu},
journal={arXiv: Combinatorics},
year={2018}
}
• Published 16 November 2018
• Mathematics
• arXiv: Combinatorics
The extremal number $\mathrm{ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [1,2]$ is realisable if there exists a graph $F$ with $\mathrm{ex}(n , F) = \Theta(n^r)$. Several decades ago, Erd\H{o}s and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $0,1, \frac{7}{5}, 2$, and the numbers of the form $1+\frac{1}{m}$, $2… 7 Citations ## Figures from this paper Many Turan exponents via subdivisions • Mathematics • 2019 Given a graph$H$and a positive integer$n$, the {\it Turan number}$\ex(n,H)$is the maximum number of edges in an$n$-vertex graph that does not contain$H$as a subgraph. A real number Turán number of bipartite graphs with no 𝐾_{𝑡,𝑡} • Mathematics • 2019 The extremal number of a graph$H$, denoted by$\mbox{ex}(n,H)$, is the maximum number of edges in a graph on$n$vertices that does not contain$H$. The celebrated Kővari-Sos-Turan theorem says that Turán Numbers of Bipartite Subdivisions • Mathematics SIAM J. Discret. Math. • 2020 The main results yield infinitely many new so-called Turan exponents: rationals$r\in (1,2)$for which there exists a bipartite graph$H$with$ex(n, H)=\Theta(n^r)$, adding to the lists recently obtained by Jiang, Ma, Yepremyan, by Kang, Kim, Liu, and by Conlon, Janzer, Lee. Negligible obstructions and Tur\'an exponents • Mathematics • 2020 We show that for every rational number$r \in (1,2)$of the form$2 - a/b$, where$a, b \in \mathbb{N}^+$satisfy$\lfloor a/b \rfloor^3 \le a \le b / (\lfloor b/a \rfloor +1) + 1$, there exists a More on the Extremal Number of Subdivisions • Mathematics Combinatorica • 2021 Given a graph H, the extremal number 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 The Extremal Number of the Subdivisions of the Complete Bipartite Graph It is proved that for a graph F, the k-subdivision of F is the graph obtained by replacing the edges of F with internally vertex-disjoint paths of length k. Some remarks on the Zarankiewicz problem The Zarankiewicz problem asks for an estimate on z(m,n;s,t), the largest number of 1's in an m×n matrix with all entries 0 or 1 containing no s×t submatrix consisting entirely of 1's. We show that a ## References SHOWING 1-10 OF 23 REFERENCES Rational exponents in extremal graph theory • Mathematics • 2015 Given a family of graphs$\mathcal{H}$, the extremal number$\textrm{ex}(n, \mathcal{H})$is the largest$m$for which there exists a graph with$n$vertices and$m$edges containing no graph from Rational exponents for hypergraph Turan problems Given a family of$k$-hypergraphs$\mathcal{F}$,$ex(n,\mathcal{F})$is the maximum number of edges a$k$-hypergraph can have, knowing that said hypergraph has$n$vertices but contains no copy of On the Extremal Number of Subdivisions • Mathematics International Mathematics Research Notices • 2019 One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich, and Sudakov, saying that if$H$is a bipartite graph with Tur\'an number of theta graphs • Mathematics • 2018 The theta graph$\Theta_{\ell,t}$consists of two vertices joined by$t$vertex-disjoint paths of length$\ell$each. For fixed odd$\ell$and large$t$, we show that the largest graph not containing Improved Bounds for the Extremal Number of Subdivisions This paper proves that there exists a constant$C'$such that$\text{ex}(n,H_t)\leq C'n^{3/2-\frac{1}{4t-6}}\$.
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