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Scope and notation

A rational number $\alpha\in[1,2)$ is realizable if one finite bipartite graph $G$ satisfies $\operatorname{ex}(n,G)=\Theta(n^\alpha)$. The results below are historical progress toward #571. They have precise statements and proof sketches or pointers here; their complete proofs are not reconstructed on this page. The 2026 resolution has its own complete proof.

Bukh–Conlon: finite forbidden families

Theorem 1.1 of Bukh–Conlon says that, for every rational $1<\alpha<2$, there is a finite family $\mathcal H$ of graphs such that $\operatorname{ex}(n,\mathcal H)=\Theta(n^\alpha)$. Here a graph counted by $\operatorname{ex}(n,\mathcal H)$ must avoid every member of the family. This does not by itself give the single forbidden graph in #571.

The PDF used for these labels is arXiv:1506.06406v2, 19 September 2017, pp. 2–3; the work appeared in JEMS 20 (2018), 1747–1757. For a rooted tree $T$ with independent roots and a nonempty internal set, the paper uses a family $\mathcal T^p$ of unions of $p$ distinct labeled copies agreeing on their roots. Their internal images may overlap. This differs from the single graph $F^{(t)}$ whose internal layers are disjoint in the present construction.

Its Lemma 1.1 gives $\operatorname{ex}(n,\mathcal T^p)=O_p(n^{2-1/\rho_T})$ when the root set is nonempty; Lemma 1.2 gives a matching lower bound for some sufficiently large $p$ when $T$ is balanced. Here $\rho_T=e(T)/|V(T)\setminus R|$, and balance means that no nonempty subset of internal vertices has smaller incident-edge density. Appropriate balanced trees then give Theorem 1.1. The upper estimate counts rooted tree embeddings; the lower estimate constructs a random polynomial graph and controls how many embeddings can share a root tuple. See §2.2 for the balanced trees and §2.3 for the lower-bound proof. Only the statements and introductory explanation on pp. 2–3 were checked here, not those complete proofs. The current arXiv record still selects v2.

Kang–Kim–Liu: single graphs and the general lower input

Theorem 1.4 of Kang–Kim–Liu states that $2-a/b$ is realizable whenever $a,b$ are positive integers, $b>a$, and $b\equiv1$ or $-1\pmod a$. The labels here refer to arXiv:1811.06916v1, 16 November 2018, p. 2; the published work is JCTB 148 (2021), 149–172. Its upper-bound constructions and the deduction of Theorem 1.4 are in §§3–4. That proof is not reproduced here.

The same source's Lemma 2.3, pp. 4–5, states the useful general lower bound: if a rooted bipartite graph $(F,R)$ is balanced with positive density $\rho_F$, then some $\ell_0$ satisfies

$$\operatorname{ex}(n,F^{(\ell)}) =\Omega(n^{2-1/\rho_F})\qquad(\ell>\ell_0).$$

Here $\rho_F$ is the number of edges touching an internal vertex divided by the number of internal vertices; balance is the corresponding lower bound for every nonempty internal subset. Its roots form a proper subset and are not required to be independent. The authors explain that the Bukh–Conlon lower-bound proof, although stated there for trees, does not use that restriction. They remove edges inside $R$ to cover adjacent roots: balance and internal incident density are unchanged, and avoidance of the smaller rooted power gives a lower bound for avoidance of the original one. These statements and that explanation were checked against pp. 4–5. The imported lower proof itself was not reread in full.

This establishes an earlier source for the general rooted lower-bound framework. The present Proposition 2.1 supplies a full reconstruction of its own generic-coefficient and ultrafilter route, rather than relying on that historical proof pointer. Both use random polynomial graphs and balance; the compiled route needs no Lang–Weil estimate. This comparison does not establish priority for individual ideas in the 2026 proof.

Kang–Kim–Liu also prove in Theorem 1.7 that their subdivision conjecture would imply the rational-exponents conjecture. That conjecture asks whether $\operatorname{ex}(n,F)=O(n^{1+\alpha})$ for a bipartite $F$ and some $\alpha>0$ implies $\operatorname{ex}(n,\operatorname{sub}(F))=O(n^{1+\alpha/2})$ for its one-subdivision. The 2026 operation adds two hubs as well as replacing edges by paths; its theorem does not prove that conjecture for arbitrary subdivisions. The current arXiv record lists only v1; equivalence to the published text was not checked here.

Jiang–Qiu: many subdivision exponents

For positive integers $p,k,b$ with $k\ge b$, Theorem 1.2 of Jiang–Qiu realizes $1+p/(kp+b)$. Its Theorem 1.3 consequently realizes $1+p/q$ for positive integers $p,q$ with $q>p^2$. These labels and statements were checked on pp. 1–2 of arXiv:1908.02385v1, 6 August 2019. Publication was in CPC 32 (2023), 134–150; only v1 appears in the current arXiv record. No comparison with the published proof was made here.

The source obtains the upper bounds from uneven subdivisions of complete bipartite graphs, paired with the rooted lower-bound framework. Its Theorem 1.10 formulates the relevant subdivision bound, §3 proves the technical Theorem 1.12, and §4 gives the deductions. Admissible-path counting and rooted embedding arguments connect this approach to the path counting used in the 2026 construction. The introductory statements and proof organization were inspected; the full technical proofs remain outside this page's scope.

Conlon–Janzer: exponents near two

Theorem 1.2 of Conlon–Janzer realizes $2-a/b$ for positive integers $a,b$ with $b\ge\max(a,(a-1)^2)$. This includes equality in the displayed condition. The edition read for the Conlon–Janzer card is arXiv:2203.03375v2, 13 December 2022, in the journal layout headed Advances in Combinatorics 2022:9, 10 pp., DOI 10.19086/aic.2022.9. The current arXiv record selects this version and gives that journal reference. The publisher's separately served file was not compared byte for byte.

Theorem 1.5 bounds the $t$-th rooted power of a height-two tree with $r$ branches, $s$ leaves at each branch, and those leaves as roots:

$$\operatorname{ex}(n,F_{r,s}^{(t)}) =O\!\left(n^{2-(r+1)/(rs+r)}\right), \qquad r\ge s+2\ge3,\quad t\ge1.$$

The tree has density $(rs+r)/(r+1)$ and is balanced for $s\le r$. The rooted lower bound, quoted as Lemma 1.3, matches this estimate for large $t$. Counting copies after controlling stars with large common neighborhoods gives the upper bound in §2; p. 3 deduces Theorem 1.2 using the earlier denominator-shifting result of Kang–Kim–Liu. Statements and this proof organization were checked on pp. 2–3. The complete counting proof and the imported denominator shift were not checked here.

Jiang–Longbrake–Yepremyan: July 2026 progress

The primary preprint Rational exponents near $3/2$, by Tao Jiang, Sean Longbrake and Liana Yepremyan, is arXiv:2607.19607v1, submitted 21 July 2026; the PDF is dated 23 July. Theorem 1.7, p. 3, states that for positive integers $\ell,r,t$ with $t\ge2$ and $r\ge2t+3$,

$$\operatorname{ex}(n,H_{r,t}^{(\ell)}) =O\!\left(n^{1+(rt-1)/(2rt+2r)}\right).$$

Here $H_{r,t}^{(\ell)}$ is the rooted power, with the old leaves as roots, of the one-subdivision of the height-two tree having $r$ first branches and $t$ leaves at each. Equivalently it is the one-subdivision of that tree's rooted power. Together with the quoted rooted lower bound for large $\ell$, the source realizes

$$1+\frac{rt-1}{2r(t+1)} =\frac32-\frac{r+1}{2r(t+1)}.$$

All three introductory pages were visually inspected, including this statement and the definition. The proof in §5 uses the anchored subfamilies introduced in §3 and the embedding lemmas for subdivisions proved in §4; it has not been reconstructed here. The current primary record, still lists only v1 and no journal reference. This is progress before the September resolution, not a later strengthening of it.

Related questions and remaining historical coverage

#713 asks about an asymptotic formula $cn^\alpha$ for every bipartite forbidden graph and about the rationality of that exponent. Realizing every rational by some graph with two-sided order bounds, as in #571, does not settle those universal or leading-constant questions.

Jiang–Longbrake's Induced rational exponents near two, v2 of 4 May 2026, concerns the maximum size of a graph with neither an induced copy of a specified graph nor a $K_{s,s}$. Its abstract gives an analogous range of exponents near two. That is a different extremal function; the abstract alone was checked, and it supplies neither a strengthening nor a proof of #571 here.

The other historical ranges retained on #571 have their existing source digests and the site's citations. Their complete proofs, the original Erdős problem passages, and exhaustive priority comparisons remain uncompiled by this source unit. Historical descriptions of #571 as open, or of a range as the frontier, are source-date claims; current status is recorded separately in the dated 2026 status record.

Bears on. #571; #713.