Improved bounds for bipartite graphs of girth eight

Improve the exponent of n in either the upper or lower bound for c1n^(1+1/15) ≤ f(n^(2/3),n) ≤ c2n^(1+1/9), where f(n,m) is the maximum number of edges in an (n,m)-bipartite graph of girth at least eight.

Background

The paper describes the best known lower and upper bounds for the number of edges in bipartite graphs with part sizes approximately n2/3 and n and girth at least eight. The exponents 1/15 and 1/9 leave a substantial gap.

References

Problem 5. Improve the magnitude (exponent of n) in either the upper or the lower bound in the inequality c1n1+1/15 ≤ f (n2/3, n) ≤ c2n1+1/9, where c1, c2 are positive constants.

— Some families of graphs, hypergraphs and digraphs defined by systems of equations  (2503.07915 - Lazebnik et al., 10 Mar 2025) in Problem 5, Section 4.10