A mathematical preprint claims that every rational exponent α at least 1 can be realized in a generalized Turán counting problem. For each such α, it gives fixed graphs Hα and Fα for which ex(n,Hα,Fα)=Θ(n^α). Put simply, the claimed order of the count is n^α, up to constant factors.
The quantity ex(n,Hα,Fα) counts copies of Hα in n-vertex graphs that avoid Fα. The counted graph Hα can be connected and have diameter at most 3, meaning any two of its vertices are at most three graph steps apart.
The manuscript is an arXiv version 1 preprint dated 20 August 2026. Its analysis works with fixed graph pairs and constructed n-vertex graphs that avoid the forbidden graph.
A counting problem with a precise target
For fixed H and F, ex(n,H,F) counts copies of H in n-vertex graphs that are F-free, meaning they avoid F. The named generalized rational exponents conjecture asks whether this count can have order Θ(n^α) for every rational α at least 1.
The proof is arranged as a localization–compression–shift framework. It takes a finite-family edge construction, turns it into a generalized Turán problem, compresses the finite forbidden family into a single forbidden graph, and uses rooted pendant leaves to shift the exponent.
First, the construction concentrates triangles
For the fractional part of the argument, the input is a rational β in the open interval from 1 to 2. A finite family Bβ has count Θ(n^β), its members are connected, and every member contains P4, a path on four vertices.
Localization turns that finite-family behavior into a triangle-counting problem. The localized obstruction family A is made up of connected graphs, and every obstruction edge lies in a triangle. For sufficiently large n, there is an n-vertex A-free graph Xn with a vertex xn of degree n−1 such that every triangle contains xn; the triangle count is Ω(n^β), and the corresponding extremal count ex(n,K3,A) is Θ(n^β).
One obstruction instead of a family
Compression replaces A with a single forbidden graph F. It forms F as a disjoint union of repeated members B1 through BM of A, with at least two such members. If L is their total component order, the construction sets the relevant clique size to q=L−1. In any F-free graph there can be at most one q-clique; deleting that clique leaves a graph that avoids A.
The lower-bound construction adds a vertex-disjoint q-clique, Kq, to X and joins it to X with exactly one cross-edge cx. That cross-edge lies in no triangle. Under the localization assumptions, the construction yields a connected counting graph H, a single forbidden graph F, and a sequence of F-free host graphs with Ω(n^β) rooted embeddings of H.
In this compressed problem, ex(n,H,F)=Θ(n^β). The counting graph H is connected and has diameter at most 3, while the lower-bound sequence supplies Ω(n^β) rooted embeddings.
Lifting the exponent
Leaf attachment extends the construction from the fractional interval to nonintegral rational exponents above 1. The proof writes α=m+θ, with m=⌊α⌋ at least 1 and 0<θ<1, then chooses β=1+θ and k=m−1 at least 0, so β lies between 1 and 2 and β+k=α.
Under the lemma’s upper-bound and rooted-embedding conditions, adjoining k pendant leaves to the distinguished root r gives ex(n,H^(k),F)=Θ(n^(β+k)). The leaf operation therefore supplies an integer shift from the β exponent to the target exponent.
Together with the base construction, this is how the paper reaches its claimed statement for every rational α at least 1. The result is asymptotic: Θ(n^α) gives the growth order, while the supplied analysis does not report exact constants or finite-n thresholds.
The boundary of the claim
The paper’s structural guarantee applies to the counted graph. It says Hα can be connected with diameter at most 3, but leaves open whether, for every rational α at least 1, both Hα and Fα can be chosen connected while preserving ex(n,Hα,Fα)=Θ(n^α).
The theorem is stated only for rational exponents α at least 1, so irrational exponents are outside its stated scope. Because it is an asymptotic Θ statement, it does not provide exact finite-n extremal values in the supplied analysis.
The work is a combinatorial proof built from finite graph families and constructed n-vertex hosts, with its endpoint expressed through asymptotic graph-counting bounds. Its central unresolved question is whether both fixed graphs can be connected without losing the same Θ(n^α) growth.
Paper data and sources
Original title: On the Generalized Rational Exponents Conjecture
Authors: Jianfeng Hou, Caihong Yang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text