EKR characterization for maximal arcs in arbitrary projective planes

Prove that if a maximal arc in an arbitrary projective plane of order q is used to form the corresponding Steiner 2-design, then every largest intersecting family of blocks is canonical.

Background

The paper proves that maximal arcs in Desarguesian projective planes yield Steiner 2-designs whose largest intersecting families are precisely the canonical ones. Computational evidence also shows the same behavior for maximal arcs in all 22 known projective planes of order 16, including non-Desarguesian planes.

On the basis of this evidence, the authors conjecture that the Desarguesian result extends to maximal arcs in arbitrary projective planes. This conjecture is not proved in the paper.

References

In view of this, we conjecture that the conclusion of Theorem~\ref{thm:EKRDesarguesianArcs} also holds in non-Desarguesian projective planes.

\begin{conjecture} Let $$ be a maximal $(n,d)$-arc in an arbitrary projective plane of order~$q$. Then the largest intersecting families in the corresponding $2$-$(n,d,1)$ design are the canonical ones. \end{conjecture}

— Erdős-Ko-Rado properties of Steiner 2-designs  (2609.26607 - Adriaensen et al., 22 Sep 2026) in Introduction, immediately after the computational discussion of maximal arcs in the 22 known projective planes of order 16