Upper bounds on the 2-colorability threshold of random d-regular k-uniform hypergraphs for k≥ 3
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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≥ k_0 where k_0 is a large non-explicit constant. Establishing the same for small values of k≥ 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≥ 3, we prove that there is an explicit d_∗(k) which gives a satisfiability upper bound for both of the models. Our upper bound d_∗(k) for k≥ 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_∗(k) coincides with the satisfiability threshold of random regular k-NAE-SAT for large enough k≥ k_0 by Ding, Sly, Sun (2014).
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