Rational exponents near two

What is the maximum number of edges in an $n$-vertex graph that does not contain a given graph $H$ as a subgraph? Denote this quantity, often called the extremal number of $H$, by $\mathrm{ex}(n,H)$. The classical results of Erdős–Stone and Erdős–Simonovits state that, for every nonempty graph $H$,

$$\mathrm{ex}(n,H) = \left(1 - \frac{1}{\chi(H)-1} + o(1)\right)\binom{n}{2},$$ where $\chi(H)$ is the chromatic number of $H$. This result supplies precise asymptotics for $\mathrm{ex}(n, H)$ whenever $\chi(H) > 2$. On the other hand, when $H$ is bipartite, it reduces to the somewhat uninformative estimate $\mathrm{ex}(n,H) = o(n^2)$. Obtaining sharper estimates for specific (families of) bipartite graphs remains one of the most formidable challenges of extremal graph theory.

The following two daring conjectures concerning extremal numbers of bipartite graphs were posed by Erdős and Simonovits. The rational exponents conjecture states that, for every bipartite $H$, there exists a rational number $r_H \in [1,2)$ such that $\mathrm{ex}(n, H) = \Theta(n^{r_H})$. The “inverse” rational exponents conjecture states that every rational number $r \in [1,2)$ is realisable in the sense that there exists a bipartite graph $H_r$ such that $\mathrm{ex}(n, H_r) = \Theta(n^r)$.

In a recent breakthrough, Bukh and Conlon [J. EMS 20 (2018), 1747–1757] came close to resolving the “inverse” conjecture by constructing, for every rational number $r \in [1,2)$, a finite family $\mathcal{H}_r$ of graphs satisfying $\mathrm{ex}(n, \mathcal{H}_r) = \Theta(n^r)$. To achieve this, they refined the random algebraic method pioneered by Bukh to construct $n$-vertex graphs with $\Omega(n^r)$ edges that avoid every graph in $\mathcal{H}_r$. Conversely, an adaptation of the famous double-counting argument due to Kővári, Sós, and Turán was used to show that, for a large enough constant $C$, every graph with $C n^r$ edges must contain some member of $\mathcal{H}_r$ as a subgraph. However, since the latter argument only implies the appearance of some member of $\mathcal{H}_r$ a subgraph, the conjecture remains open in its original “single graph” form.

Since the work of Bukh and Conlon, there has been swift progress towards the single graph form of the Erdős–Simonovits conjecture. In particular, Jiang and Qiu showed that every rational number of the form $1+p/q$ with $q > p^2$ is realisable. The main result of this paper is that every rational number of the form $2 - a/b$ with $b \ge \max\{a, (a-1)^2\}$ is also realisable or, informally speaking, every rational number sufficiently close to two is a Turán exponent.