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Characterize predicates with linear non-redundancy

Determine all predicates R ⊆ D^r for which the non-redundancy NRD(R,n) grows linearly with n; equivalently, characterize the class of CSP predicates that have NRD(R,n) = Θ(n).

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Background

Non-redundancy NRD(R,n) measures the largest size of a non-redundant CSP(R) instance on n variables. While various families (e.g., Mal'tsev predicates) are known to have NRD(R,n) = O(n), a complete characterization of when the growth is linear remains elusive.

A precise criterion for linear non-redundancy would also clarify optimal sparsifiability for many CSPs, given the paper’s tight link between sparsifiability and non-redundancy.

References

For a general predicate $R \subseteq Dr$, there is no simple (even conjectured) expression for $NRD(R,n)$. In fact, even determining when $NRD(R,n) = \Theta(n)$ is an open question (e.g., ).

— Redundancy Is All You Need (2411.03451 - Brakensiek et al., 5 Nov 2024) in Subsection “Open Questions” (Introduction)