Largest hypercube subgraph in the transposition Cayley graph

Determine the largest full subgraph of the Cayley graph of the symmetric group $S_n$ generated by all transpositions that is isomorphic to a hypercube, and determine the maximum dimension of any such hypercube.

Background

The paper considers the Cayley graph Γn\Gamma_n of the symmetric group SnS_n, with vertices given by permutations and edges joining permutations that differ by multiplication by a transposition. The problem asks for the largest induced—or “full”—subgraph of this highly structured graph that is isomorphic to a hypercube, together with the largest possible hypercube dimension.

The paper explains that computational searches using FunSearch and AlphaEvolve found natural hypercubes whose dimension grows like nlog⁡nn\log n, within a constant factor of the best possible bound. However, the exact largest subgraph and exact maximum dimension are not determined in the paper, so the stated extremal question remains unresolved.

References

What is the largest full subgraph of $\Gamma_n$ that is isomorphic to a hypercube? What is the largest dimension of such a hypercube that can occur?

— AI and humanity's long conversation  (2610.13546 - Williamson, 9 Oct 2026) in Section “LLMs and mathematics,” Question environment following the definition of the Cayley graph $\Gamma_n$

For $S_n$ the question includes the exact value of $N(S_n)$. It lies between $\lfloor3n/4\rfloor$ and $n-2$ for $n\ge5$, and it equals $\lfloor3n/4\rfloor$ for every $n\le8$ (\cref{thm:solvable,thm:abelruffini}). Is $N(S_n)=\lfloor3n/4\rfloor$ for all $n$?

— Learning a Group Element from Its Action: Galois Correspondences, Optimal ERM, and an Abel-Ruffini Threshold for the Natarajan Dimension  (2610.18154 - Li, 13 Oct 2026) in Section 7, Discussion and open problems, paragraph 2