Common-neighbour conjectures for Saxl graphs; Kourovka Notebook Problem 21.29

Let $G$ be a finite primitive permutation group on a set $\Omega$ , and let $b$ be its base size. Define the generalised Saxl graph $\Sigma_b(G)$ to have vertex set $\Omega$ , with distinct vertices $x,y\in\Omega$ adjacent if and only if $\{x,y\}$ is contained in a base of $G$ of size $b$ . The common-neighbour conjecture asserts that every two distinct vertices of $\Sigma_b(G)$ have a common neighbour; equivalently, for all distinct $x,y\in\Omega$ , there exists $z\in\Omega\setminus\{x,y\}$ such that both $\{x,z\}$ and $\{y,z\}$ are contained in bases of size $b$ . For $b=2$ , this is the Burness–Giudici conjecture for the ordinary Saxl graph; for arbitrary $b$ , it is the extension formulated by Freedman, Huang, Lee and Rekvénnyi.

References

Primary source

arXiv

Additional references

Common neighbour conjectures for Saxl graphs fail at every base size — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new unrefereed paper claims counterexamples to the common-neighbour conjectures in every base size, while some special cases remain open.

The entry tracks two common-neighbour conjectures for Saxl graphs and Kourovka Notebook Problem $21.29$ . The latest paper claims a broad negative resolution, but also proves the conjecture for one specified sporadic affine class.

Known results

Verification was known for primitive groups of degree at most $4095$ .
The conjecture was established for almost simple groups with soluble point stabilisers, many sporadic cases, and most affine-type groups with sporadic stabilisers.
On December 27, 2025, separate preprints claimed the conjecture for several $\mathrm{PSU}(3,q)$ , $\mathrm{Sz}(q)$ , and $\mathrm{Ree}(q)$ families.

September 2026 claimed counterexamples

The preprint claims counterexamples for every base size $B\geq 2$ , including additional base-two families, and claims a positive result for a specified sporadic affine class. These results are unverified.

Current status (as of September 2026): A preprint claims the conjectures fail for every $B\geq 2$ , but the claim is unverified and almost simple and diagonal base-two cases remain open.

Sources

No solutions have been posted yet.