Jiang 2026 rational exponents near 3 2

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theorem_1_7: States that the l-th power, rooted at its leaves, of the once-subdivided height-two tree with r branches of t leaves has extremal number O(n^{1+(rt-1)/(2rt+2r)}) for t at least 2 and r at least 2t+3.

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T. Jiang, S. Longbrake and L. Yepremyan, Rational exponents near 3/2, arXiv:2607.19607v1 [math.CO], submitted 21 July 2026 (the title page is dated 23 July 2026), 28 pp. No journal version is known here.

The copy read for this card is the arXiv v1 PDF, 468,568 bytes: the stamp "arXiv:2607.19607v1 [math.CO] 21 Jul 2026" runs down its first page, and the printed page numbers 1--28 agree with the PDF pages. It has a text layer, in which the statements below were read. The download URL was not recorded; the arXiv record is https://arxiv.org/abs/2607.19607v1. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2607.19607), every other right reserved.

Read status: claims checked for Theorem 1.7, whose statement and defining notation (Section 1, pp. 2--3, and Definition 2.1, pp. 3--4) were read clause by clause; no proof was read.

Contents

• Conjecture 1.1 (p. 1), the rational exponents conjecture of Erdős and Simonovits: for every rational $\gamma\in[1,2]$ there is a graph $H$ with $\operatorname{ex}(n,H)=\Theta(n^\gamma)$. The introduction records the Bukh--Conlon finite-family theorem and the single-graph ranges of Jiang--Qiu (Theorem 1.2, $\gamma=1+a/b$ with $b>a^2$) and Conlon--Janzer (Theorem 1.3, $\gamma=2-a/b$ with $b\ge\max\{a,(a-1)^2\}$; the print has "$b\geq\{a,(a-1)^2\}$" on p. 2 and the abstract $b>\max\{a,(a-1)^2\}$), pp. 1--2.
• Rooted graphs (p. 2): for a graph $F$ with root set $R$, $\rho_F(S)$ is the number of edges incident to $S$ divided by $|S|$, $\rho(F)=\rho_F(V(F)\setminus R)$, and $(F,R)$ is balanced if $\rho_F(S)\ge\rho(F)$ for every nonempty $S\subseteq V(F)\setminus R$; $F_R^\ell$ is the $\ell$-th power of $F$ rooted at $R$, $\ell$ copies of $F$ sharing the roots and disjoint elsewhere.
• Theorem 1.4 (p. 2; Bukh--Conlon, quoted): for every balanced rooted bipartite graph $(F,R)$ with $\rho(F)>0$ there is $\ell_0$ such that $\operatorname{ex}(n,F_R^\ell)=\Omega(n^{2-1/\rho(F)})$ for all $\ell\ge\ell_0$. Conjecture 1.5 (p. 2), the Bukh--Conlon conjecture, asks for the matching upper bound $O_\ell(n^{2-1/\rho(T)})$ for every balanced rooted tree $(T,R)$ and every $\ell$; p. 2 notes that it implies Conjecture 1.1.
• Theorem 1.6 (p. 3; Conlon--Janzer, quoted): for the height-two tree $T_{r,t}$ (an $r$-star with $t$ leaves joined to each of its leaves) rooted at its leaves, $\operatorname{ex}(n,F_{r,t}^\ell)=O(n^{2-(r+1)/(rt+r)})$ when $r\ge t+2\ge3$.
• Theorem 1.7 (p. 3), the main theorem: for positive integers $\ell,r,t$ with $t\ge2$ and $r\ge2t+3$, the $\ell$-th power $H_{r,t}^\ell$ of the once-subdivided tree $T'_{r,t}$, rooted at its leaves, has $\operatorname{ex}(n,H_{r,t}^\ell)=O(n^{1+(rt-1)/(2rt+2r)})$. Page 3 says this verifies the Bukh--Conlon conjecture for these subdivided trees, and the abstract states the resulting exponents $\gamma=1+(rt-1)/(2rt+2r)$ as cases of Conjecture 1.1; the case $t=1$ is attributed to the main theorem of Janzer's paper [12].
• Section 6 (p. 27): the theorem also proves the Kang--Kim--Liu conjecture ($\operatorname{ex}(n,H)=O(n^{1+\alpha})$ for a bipartite $H$ should give $\operatorname{ex}(n,H')=O(n^{1+\alpha/2})$ for the once-subdivision $H'$) for rooted powers of $T_{r,t}$ in the same range; the authors think the method likely to give the Bukh--Conlon conjecture for the $p$-subdivisions of $T_{r,t}$ for every even $p$ when $r$ is moderately large compared to $t$.

Compiled scope

Pages 1--3 and the top of p. 4 (abstract, introduction and Definition 2.1) and the concluding remarks on p. 27 were read in the text layer. Sections 2--5 (pp. 3--26), which develop the anchored-subfamily and embedding lemmas and prove Theorem 1.7 in Section 5 (pp. 17--26), were not read beyond their headings. Nothing here is independently reviewed.

Bears on. #571, as the [JLY26] row of that page's exponent table: Theorem 1.7 gives the upper bound $O(n^{1+(rt-1)/(2rt+2r)})$, and the abstract states that the paper establishes the rational exponents conjecture for $\gamma=1+(rt-1)/(2rt+2r)$, $t\ge2$, $r\ge2t+3$, which equals the row's $3/2-(r+1)/(2r(t+1))$. The matching lower bound comes from the quoted Theorem 1.4 for large $\ell$, as p. 2 explains; the balance this needs was not checked here.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.