Exact Helly numbers for types D and E6

Determine the exact Helly numbers for connected almost simple complex algebraic groups of type D_m for m 4 and of type E_6, resolving whether h(D_m)=m+1 or m+2 and whether h(E_6)=7 or 8.

Background

The paper establishes the general upper bound h(G) m+2 for an almost simple group of semisimple rank m. It proves the lower bound h(D_m)m+1 through the embedding GL_mSO_{2m}, leaving h(D_m) equal to either m+1 or m+2. For type E_6, the maximal-rank inclusion A_5A_1E_6 gives the lower bound 7, while the general rank bound gives the upper bound 8.

The exact Helly number h(G) is equivalent to the apartment Helly number of the spherical building associated with G. Resolving these two cases would therefore complete the determination of the corresponding Helly numbers for all almost simple types treated in the paper.

References

In particular, the exact values and bounds for h(\Delta_{\mathrm{sph}(G)}) coincide with those listed in Corollary~\ref{cor:helly-values-by-type}, leaving open cases only for types D_m and E_6.

— Helly Numbers for Connected Reductive Groups and Splitting of Toric Principal Bundles  (2610.11340 - Tsvelikhovskiy, 8 Oct 2026) in Corollary (Section 6, subsection "Spherical buildings and common maximal tori"); see also Corollary (Section 4, subsection "Helly's Number for Connected Reductive Groups")