Graph-theoretic formulation of the m-ary sensitivity conjecture

Establish that for every positive integer m there exists μ > 0 such that for any collection of (possibly empty) induced subgraphs H_1, H_ε, ..., H_{ε^{m−1}} of the Hamming graph H(n,m) whose vertex sets partition the vertex set and satisfy ∑_{k=0}^{m−1} |ρ(V(H_{ε^k}))| ε^k ≠ 0, the quantity (m−1)n − min{δ(H_{ε^k}) : 0 ≤ k ≤ m−1} grows polynomially in n, i.e., belongs to Ω(n^μ).

Background

Using a generalized equivalence theorem, the authors translate the m-ary sensitivity conjecture into a statement about imbalanced partitions of the Hamming graph H(n,m) into induced subgraphs with controlled minimum degree. Here δ denotes minimum degree, and ρ is the rotation of the partition aligned with the canonical m-coloring of H(n,m).

When m is prime, the imbalance condition reduces to requiring that not all rotated color classes have equal size; this emphasizes the graph-theoretic nature of the conjecture and connects it to defective colorings and induced subgraph degree bounds.