On the Cameron-Praeger conjecture for Steiner 6-designs

@misc{wang2026cameronpraeger,
  title = {{On the Cameron-Praeger conjecture for Steiner 6-designs}},
  author = {Ximin Wang and Zheng Huang and Weijun Liu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06081},
  eprint = {2611.06081},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06081},
}

TY  - UNPB
TI  - On the Cameron-Praeger conjecture for Steiner 6-designs
AU  - Ximin Wang
AU  - Zheng Huang
AU  - Weijun Liu
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06081
UR  - https://arxiv.org/abs/2611.06081
AB  - A long-standing and far-reaching conjecture of Cameron and Praeger (1993) asserts that no non-trivial block-transitive $6−(v,k,λ)$ designs exist. Huber (2010) reduced the Steiner case (λ=1) to a single infinite family of possible exceptions, where the automorphism group is G=PGaL(2,p^e) acting on the points of the projective line, with p∈{2,3} and e an odd prime power. In this paper, we resolve this final case. By combining orbit-stabiliser analysis for the action of specific involutions in PGaL(2,p^e) with the modular arithmetic constraints derived from the design parameters, we prove that no such Steiner 6-designs can exist for any admissible q. This completes the proof that the Cameron-Praeger conjecture is true for the important case of non-trivial Steiner 6-designs.
ER  -