Towards infinite PCSP: a dichotomy for monochromatic cliques

TL;DR

This paper introduces Promise MMSNP, extends the MMSNP-CSP correspondence to promises, and classifies the complexity of promise graph coloring problems under a conjecture, with implications for hypergraph coloring.

Contribution

It establishes a poly-time equivalence between PMMSNP and Promise CSPs and provides a complexity classification for promise monochromatic clique problems.

Findings

PMMSNP problems are poly-time equivalent to Promise CSPs.

Classifies promise monochromatic clique problems' complexity under the Rich 2-to-1 Conjecture.

Proves NP-hardness of hypergraph coloring with reconfigurability constraints.

Abstract

The logic MMSNP is a well-studied fragment of Existential Second-Order logic that, from a computational perspective, captures finite-domain Constraint Satisfaction Problems (CSPs) modulo polynomial-time reductions. At the same time, MMSNP contains many problems that are expressible as $\omega$-categorical CSPs but not as finite-domain ones. We initiate the study of Promise MMSNP (PMMSNP), a promise analogue of MMSNP. We show that every PMMSNP problem is poly-time equivalent to a (finite-domain) Promise CSP (PCSP), thereby extending the classical MMSNP-CSP correspondence to the promise setting. We then investigate the complexity of PMMSNPs arising from forbidding monochromatic cliques, a class encompassing promise graph colouring problems. For this class, we obtain a full complexity classification conditional on the Rich 2-to-1 Conjecture, a recently proposed perfect-completeness surrogate of the Unique Games Conjecture. As a key intermediate step which may be of independent interest, we prove that it is NP-hard, under the Rich 2-to-1 Conjecture, to properly colour a uniform hypergraph even if it is promised to admit a colouring satisfying a certain technical condition called reconfigurability. This proof is an extension of the recent work of Braverman, Khot, Lifshitz and Minzer (Adv. Math. 2025). To illustrate the broad applicability of this theorem, we show that it implies most of the linearly-ordered colouring conjecture of Barto, Battistelli, and Berg (STACS 2021).