Infinite families of designs with non-design non-canonical maximum cliques

Determine whether there exists an infinite family of Steiner 2-designs whose block graphs have non-canonical maximum cliques without an associated design structure.

Background

For a Steiner 2-design, maximum intersecting families of blocks correspond to maximum cliques in the block graph. Canonical maximum cliques consist of all blocks through a fixed point, whereas non-canonical maximum cliques need not arise from a smaller subdesign or its dual.

The paper explains that known examples include affine-plane constructions and certain isolated examples, but the existence of an infinite family of designs with non-canonical maximum cliques lacking any design structure remains unresolved. The paper later states that its computational examples provide strong counterexamples to the related subdesign question, but do not resolve this infinite-family problem.

References

If the problem is interpreted in a way that includes isomorphism and duality, then it can still be considered open: so far there have been no ``strong counterexamples''. Barring affine planes, the following problem is open in either sense.

Does there exist an infinite family of $2$-designs whose block graphs have non-canonical maximum cliques without a design structure?

— Erdős-Ko-Rado properties of Steiner 2-designs  (2609.26607 - Adriaensen et al., 22 Sep 2026) in Introduction, Problem 2 (citing Problem 2 of Goryainov and Konstantinova)