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Without interactions this is a textbook calculation; with them there is no accepted proof.","posed_since":"","precise":"$N$ bosons in a box of side $L$ in $R^{3}$, $H = \\sum_i -\\operatorname{Lap}_i + \\sum_{i<j} v(x_i - x_j)$ with $v \\ge 0$ compactly supported, density $\\rho = N/L^{3}$ fixed, $L \\to \\infty$. Prove at $T = 0$ or at small $T > 0$ that the one-body density matrix has off-diagonal long-range order, lim_{|x-y| -> infinity} gamma^(1)(x,y) = $\\rho$_0 > 0. Answer: a proof for some open range of rho and $T$.","problem_ref":"281467524de81f66573dcbbed47192e3c0423267cd2a96b34feb574cb93de781","references":"","settled_by":"A proof of a nonzero condensate density in the thermodynamic limit for one continuum model.","status_note":"Proven in the Gross-Pitaevskii scaling (Lieb and Seiringer, 2002) and for lattice hard-core bosons at half filling (Dyson, Lieb and Simon, 1978), not for continuum gases in the thermodynamic limit; a preprint by Suto (arXiv 1710.04441, 2017) claims a proof via Feynman-Kac permutation cycles and lists no journal publication.","title":"Condensation of a repulsive Bose gas in the thermodynamic 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