[Paper Review] On the rational Tur\'an exponents conjecture

Dong Yeap Kang, Jaehoon Kim+1 more|arXiv (Cornell University)|Nov 16, 2018

Limits and Structures in Graph Theory3 references4 citations

TL;DR

This paper makes significant progress on the rational Turán exponents conjecture by proving that $2 - \frac{a}{b}$ is realisable for all integers $a,b \geq 1$ with $b > a$ and $b \equiv \pm 1 \pmod{a}$, thereby establishing infinitely many new realisable exponents and confirming $2 - \frac{1}{m}$ as limit points. It further proposes a subdivision conjecture implying that all rational numbers in $[1,2]$ are realisable, linking extremal graph theory to structural graph properties.

ABSTRACT

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 $1+\\frac{1}{m}$, $2-\\frac{1}{m}$, $2-\\frac{2}{m}$ for integers $m \\geq 1$. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than two numbers $1$ and $2$. In this paper, we make progress on the conjecture of Erd\\H{o}s and Simonovits. First, we show that $2 - \\frac{a}{b}$ is realisable for any integers $a,b \\geq 1$ with $b>a$ and $b \\equiv \\pm 1 ~({\ m mod}\\:a)$. This includes all previously known ones, and gives infinitely many limit points $2-\\frac{1}{m}$ in the set of all realisable numbers as a consequence. Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.

Motivation & Objective

• To resolve the long-standing rational Turán exponents conjecture by identifying new realisable exponents in the interval [1,2].
• To establish that $2 - \frac{a}{b}$ is realisable for integers $a,b \geq 1$ with $b > a$ and $b \equiv \pm 1 \pmod{a}$, extending known results.
• To propose a novel subdivision conjecture on bipartite graphs that, if true, would imply the full rational Turán exponents conjecture.
• To provide structural insights into extremal graphs via blow-ups and subdivisions, particularly for trees and complete bipartite graphs.
• To identify new limit points in the set of realisable exponents, including $2 - \frac{1}{m}$ for all $m \in \mathbb{N}$.

Proposed method

• Uses a novel construction based on blow-ups of rooted trees and their subdivisions to generate extremal graphs with controlled density.
• Applies a variant of the dependent random choice method to control neighborhood growth and ensure the existence of dense subgraphs avoiding forbidden configurations.
• Employs Hall’s theorem and matching arguments in bipartite graphs to embed specific subgraphs like $\mathrm{sub}(K_{s,t})$ and $T_{s,t}^\ell$.
• Introduces the concept of $D_{s,t}^\ell$, the $\ell$-blow-up of a tree $D_{s,t}$, to analyze extremal numbers via recursive neighborhood expansion.
• Leverages known extremal bounds and reduction theorems (e.g., Faudree-Simonovits) to derive upper bounds on extremal numbers of subdivided graphs.
• Proposes a conjecture on the extremal number of $1$-subdivisions of bipartite graphs, suggesting a power-law scaling that preserves the exponent structure under subdivision.

Experimental results

Research questions

• RQ1 Are there infinitely many new realisable exponents in the interval $[1,2]$ beyond the known $1 + \frac{1}{m}$, $2 - \frac{1}{m}$, and $2 - \frac{2}{m}$?
• RQ2 Can the condition $b \equiv \pm 1 \pmod{a}$ be used to systematically generate realisable exponents of the form $2 - \frac{a}{b}$?
• RQ3 Does the $1$-subdivision of a bipartite graph $F$ with $\mathrm{ex}(n,F) = O(n^{1+\alpha})$ satisfy $\mathrm{ex}(n,\mathrm{sub}(F)) = O(n^{1+\alpha/2})$?
• RQ4 Is the extremal number of the $\ell$-blow-up of a tree $H_{t,s}$ asymptotically $\Theta(n^{1 + \frac{t}{1 + (s+1)t}})$ for large $\ell$?
• RQ5 What is the threshold $\ell_0$ beyond which the extremal number of $D_{t-1,s-1}^\ell$ transitions from $O(n^{4/3})$ to $O(n^{2 - \frac{t}{st-1}})$?

Key findings

• The number $2 - \frac{a}{b}$ is realisable for all integers $a,b \geq 1$ with $b > a$ and $b \equiv \pm 1 \pmod{a}$, significantly expanding the set of known realisable exponents.
• As a consequence, $2 - \frac{1}{m}$ is a limit point in the set of realisable exponents for every $m \in \mathbb{N}$, establishing infinitely many such limit points.
• The proposed subdivision conjecture implies that every rational number in $[1,2]$ is realisable, should it be true.
• For the $1$-subdivision of $K_{s,t}$ with $t \geq s$, the extremal number satisfies $\mathrm{ex}(n,\mathrm{sub}(K_{s,t})) \leq O(n^{\frac{3}{2} - \frac{1}{4s-2}})$, improving prior bounds.
• The extremal number of $D_{2,1}^2$ is $\Theta(n^{4/3})$, while for large $\ell$, $\mathrm{ex}(n,D_{2,1}^\ell) = \Theta(n^{7/5})$, indicating a phase transition at $\ell = 3$, not $\ell = 2$.
• The extremal number of $T_{4,7}^\ell$ is $\Theta(n^{10/7})$ for sufficiently large $\ell$, confirming a new realisable exponent via blow-up construction.