Strong m-ary Sensitivity Conjecture

Establish that for any integer m ≥ 2 there exists μ > 0 such that for any partition of the Hamming graph H(n,m) into (possibly empty) induced subgraphs H_1, ..., H_m with ∑_{i=1}^{m} |V(H_i)| ε^i ≠ 0, the maximum degree among the parts satisfies max{Δ(H_1), ..., Δ(H_m)} ∈ Ω(n^μ).

Background

The authors propose a stronger and more natural graph-theoretic form of the m-ary sensitivity conjecture. Rather than bounding a minimum-degree deficit via the equivalence theorem, this statement asserts that in any sufficiently imbalanced partition of H(n,m) into m induced subgraphs, at least one part must have polynomially large maximum degree.

This formulation highlights the connection to defective colorings while introducing novel imbalance requirements on color classes, and it would imply the graph-theoretic formulation of the m-ary sensitivity conjecture.

References

We present a stronger conjecture in more natural terms: Conjecture [Strong m-ary Sensitivity Conjecture] Let m be a positive integer and \varepsilon an m-th primitive root of unity. There exists \mu > 0 such that for any partition of H(n,m) into (possibly empty) induced subgraphs H_1, \dots,H_{m} with \sum_{i=1}{m}|V(H_{i})|\ \varepsiloni\neq 0, we have max{\Delta(H_{1}),\dots,\Delta(H_{m})} \in \Omega(n{\mu})\.

— Sensitivity of $m$-ary functions and low degree partitions of Hamming graphs (2409.16141 - Asensio et al., 24 Sep 2024) in Section 6: Conclusions (Conjecture \ref{conj:stronger})