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The coefficient is known only from computer simulations.","posed_since":"","precise":"For the dilute 3D Bose gas, prove that $(T_{c} - T_{c}^{0})/T_{c}^{0} = c\\rho^{1/3}a + o(\\rho^{1/3}a)$, with $T_{c}^{0}$ the ideal-gas value, and determine c (lattice Monte Carlo gives c approximately 1.3, Arnold and Moore; Kashurnikov, Prokof'ev and Svistunov, 2001). Answer: the existence of the phase transition and the value of c.","problem_ref":null,"references":"","settled_by":"A proof of a phase transition at $T_{c}$ with rigorous bounds on $c$.","status_note":"","title":"Shift of the condensation temperature caused by weak 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