Upper bounds on the $2$-colorability threshold of random $d$-regular $k$-uniform hypergraphs for $k\geq 3$

TL;DR

This paper establishes explicit upper bounds on the 2-colorability threshold for random d-regular k-uniform hypergraphs with k≥3, matching physics predictions and connecting to known thresholds for NAE-SAT, advancing understanding of phase transitions in random CSPs.

Contribution

It provides the first explicit upper bounds for the 2-colorability threshold for all k≥3, aligning with physics predictions and known thresholds for large k.

Findings

Derived explicit upper bounds d_*(k) for 2-colorability for all k≥3.

Showed d_*(k) matches physics predictions for hypergraph 2-coloring.

Connected the bounds to the satisfiability threshold of regular k-NAE-SAT for large k.

Abstract

For a large class of random constraint satisfaction problems (CSP), deep but non-rigorous theory from statistical physics predict the location of the sharp satisfiability transition. The works of Ding, Sly, Sun (2014, 2016) and Coja-Oghlan, Panagiotou (2014) established the satisfiability threshold for random regular $k$-NAE-SAT, random $k$-SAT, and random regular $k$-SAT for large enough $k\geq k_0$ where $k_0$ is a large non-explicit constant. Establishing the same for small values of $k\geq 3$ remains an important open problem in the study of random CSPs. In this work, we study two closely related models of random CSPs, namely the $2$-coloring on random $d$-regular $k$-uniform hypergraphs and the random $d$-regular $k$-NAE-SAT model. For every $k\geq 3$, we prove that there is an explicit $d_{\ast}(k)$ which gives a satisfiability upper bound for both of the models. Our upper bound $d_{\ast}(k)$ for $k\geq 3$ matches the prediction from statistical physics for the hypergraph $2$-coloring by Dall'Asta, Ramezanpour, Zecchina (2008), thus conjectured to be sharp. Moreover, $d_{\ast}(k)$ coincides with the satisfiability threshold of random regular $k$-NAE-SAT for large enough $k\geq k_0$ by Ding, Sly, Sun (2014).