VC-dimension–degree trade-offs on Boolean slices

Establish a trade-off relationship between the VC-dimension and harmonic degree of Boolean functions defined on the k-th slice \(\binom{[n]}{k}\) of the Boolean cube.

Background

The paper proves that every non-zero Boolean function on the full Boolean cube satisfies trade-offs between VC-dimension and both real polynomial degree and $\mathbb{F}_2$-algebraic degree. It asks whether an analogous result holds for Boolean functions restricted to a fixed slice $\binom{[n]}{k}$.

On a slice, every real-valued function has a unique harmonic multilinear polynomial representation of degree at most $\min(k,n-k)$, so harmonic degree $\deg_H(f)$ is the natural algebraic complexity measure. The open problem is to relate this degree to the VC-dimension of the $k$-uniform hypergraph represented by the Boolean function.

The same subsection notes that even the slice analogue of the Sauer–Shelah–Perles extremal problem remains unresolved: determining the largest family of subsets in the slice with a prescribed VC-dimension. This provides additional motivation for seeking a VC-dimension–degree inequality on slices.