Finiteness of biplanes

Determine whether only finitely many biplanes exist.

Background

Equality in the paper’s maximum-degree bound n≤binom(d,2)+1 occurs precisely for incidence graphs of biplanes, namely symmetric block designs with parameters (v,k,λ)=(1+binom(d,2),d,2). Such designs yield dHp bigraphs attaining the lower bound exactly.

The paper notes that only finitely many biplanes are currently known and explicitly states the conjecture that the total number of biplanes is finite. This issue determines whether constructions attaining the degree lower bound can exist for infinitely many parameter values.

References

However, it is conjectured that only finitely many biplanes exist, which means that perhaps we cannot hope for a general construction of this type.

— Bipartite graphs with the double Hall property  (2502.10903 - Chen et al., 15 Feb 2025) in Section 5, subsection “Constructions via biplanes”