8-cycle-free BΓ3 graphs

Construct a graph BΓ3(Fq; f2, f3, f4) that contains no 8-cycles for infinitely many prime powers q, thereby potentially establishing the order of magnitude ex(ν,{C8})=Θ(ν5/4).

Background

The paper discusses the long-standing problem of determining the magnitude of the Turán number for graphs without 8-cycles. The known lower and upper bounds are Ω(ν6/5) and O(ν5/4), respectively. A positive answer would close this gap and yield the conjectured exponent.

References

Problem 1. Is there a graph BΓ3(Fq; f2, f3, f4) that contains no 8-cycles for infinitely many q?

— Some families of graphs, hypergraphs and digraphs defined by systems of equations  (2503.07915 - Lazebnik et al., 10 Mar 2025) in Problem 1, Section 4.3

The first unresolved case is $k=4$; see . The general lower bound then has exponent $6/5$, whereas the conjectured exponent is $5/4$.

— Balanced even cycles in signed graphs:Turán bounds, double covers, and parity obstructions  (2609.10975 - Wang et al., 10 Sep 2026) in Section 1, subsection “Balanced even cycles”