First open case: strong ternary sensitivity on H(n,3)

Establish that there exists μ > 0 such that for any partition of the Hamming graph H(n,3) into three (possibly empty) induced subgraphs H_1, H_2, H_3 that are not all of the same order, the maximum degree among the parts satisfies max{Δ(H_1), Δ(H_2), Δ(H_3)} ∈ Ω(n^μ).

Background

As a tempting first nontrivial instance of the strong m-ary sensitivity conjecture, the authors single out the case m = 3. For prime m=3, the imbalance condition simplifies to the parts not all having equal order, making this a natural and concrete target for progress.

Resolving this case would provide evidence toward the general strong conjecture and illuminate the structure of low-degree partitions in Hamming graphs.