# 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} }

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

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

- MathematicsSIAM 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

- MathematicsCombinatorica
- 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

- MathematicsSIAM J. Discret. Math.
- 2020

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

- Mathematics
- 2020

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…

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