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Desarguesianity of finite cyclic projective planes

Show that every finite cyclic projective plane is Desarguesian.

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Background

Desargues’s theorem characterizes a strong geometric property; projective planes arising from finite fields are Desarguesian. While non-Desarguesian planes exist, the conjecture asserts that the additional cyclic symmetry forces Desarguesianity.

The paper notes this conjecture is open and discusses its potential implications for perfect difference sets.

References

A related conjecture is the following: \begin{conjecture} \label{conjecture:desargues} Every finite cyclic projective plane (and thus perfect difference set) is Desarguesian. \end{conjecture} … However, because Conjecture~\ref{conjecture:desargues} is open, we will not mention Desargues’s theorem in the remainder of this work.

— Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof (2510.19804 - Alexeev et al., 22 Oct 2025) in Conjecture (label: conjecture:desargues), Section 4 (Cyclic projective planes)