Clique number of the complete-graph spanning-tree t-disjointness graph

Determine the clique number ω(Γ_t(K_n)) of the spanning tree t-disjointness graph of the complete graph K_n.

Background

The paper notes that Nash-Williams’ theorem determines the clique number of the ordinary spanning tree disjointness graph Γ(G), corresponding to t=1, for arbitrary G. However, for the generalized t-disjointness graph Γ_t(K_n), the clique number remains unresolved, even in the highly symmetric complete-graph setting.

References

In fact, the clique number of $\omega(\Gamma_t(K_n))$ is open.

— Intersecting Families of Spanning Trees  (2502.08128 - Frankl et al., 12 Feb 2025) in Section 7, Open Problems; Subsection 7.2, Other Directions