Rational Turán exponents of a single k-uniform hypergraph
Abstract
For a fixed k-uniform hypergraph F, let exk(n,F) denote the maximum number of edges in an F-free k-uniform hypergraph on n vertices. We prove that for every integer k≥2 and every rational r∈[0,1] there exists a k-uniform hypergraph Fk,r such that exk(n,Fk,r)=Θ(nk−r). The case k=2 is the rational exponents conjecture of Erdős and Simonovits \cite{Erdos1988}, also known as Erdős problem \#571 \cite{BloomErdoswebsite}; a proof was announced in September 2026 and is expounded by Conlon et al.\ \cite{Conlon-Janzer-Jiang-Kim-Lee-Liu-Longbrake-Yepremyan}. For k≥3 the theorem extends their result to k-uniform hypergraphs. In the single-hypergraph setting, this strengthens Fitch's finite-family theorem \cite{Fitch} by realizing each exponent with a single forbidden k-uniform hypergraph for 0≤ r≤1. We also show, via an explicit two-edge construction, that for k≥4 this strengthening fails for ⌈ k/2⌉<r≤ k−1: no single k-uniform hypergraph F then satisfies exk(n,F)=Θ(nk−r). The main result of this paper has been formalized in Lean 4, conditional on two explicitly stated axioms.
Related
- Single-graph rational exponents in mixed multicolour Ramsey theory
- Improved Sidorenko bound for hypergraph Turan densities
- Hypergraph Turán Problem of the Generalized Triangle with Bounded Matching Number
- Vertex-colored Turán theorems with applications in extremal hypergraph problems
- Turán densities of hypergraph augmentations
BibTeX
@misc{liu2026rational,
title = {{Rational Turán exponents of a single $k$-uniform hypergraph}},
author = {Chang Liu and Jue Dai},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07290},
eprint = {2611.07290},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07290},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rational Turán exponents of a single k-uniform hypergraph
AU - Chang Liu
AU - Jue Dai
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07290
UR - https://arxiv.org/abs/2611.07290
AB - For a fixed k-uniform hypergraph F, let ex_k(n,F) denote the maximum number of edges in an F-free k-uniform hypergraph on n vertices. We prove that for every integer k≥2 and every rational r∈[0,1] there exists a k-uniform hypergraph F_k,r such that ex_k(n,F_k,r)=Θ(n^k−r). The case k=2 is the rational exponents conjecture of Erdős and Simonovits \cite{Erdos1988}, also known as Erdős problem \#571 \cite{BloomErdoswebsite}; a proof was announced in September 2026 and is expounded by Conlon et al.\ \cite{Conlon-Janzer-Jiang-Kim-Lee-Liu-Longbrake-Yepremyan}. For k≥3 the theorem extends their result to k-uniform hypergraphs. In the single-hypergraph setting, this strengthens Fitch's finite-family theorem \cite{Fitch} by realizing each exponent with a single forbidden k-uniform hypergraph for 0≤ r≤1. We also show, via an explicit two-edge construction, that for k≥4 this strengthening fails for ⌈ k/2⌉<r≤ k−1: no single k-uniform hypergraph F then satisfies ex_k(n,F)=Θ(n^k−r). The main result of this paper has been formalized in Lean 4, conditional on two explicitly stated axioms.
ER -