In this work, we investigate the existence of string theory solutions with a d-dimensional (quasi-) de Sitter spacetime, for 3 ≤ d ≤ 10. Considering classical compactifications, we derive no-go theorems valid for general d. We use them to exclude (quasi-) de Sitter solutions for d ≥ 7. In addition, such solutions are found unlikely to exist in d = 6, 5. For each no-go theorem, we further compute the d-dependent parameter c of the swampland de Sitter conjecture, $M_p\frac{\mid \nabla V\mid }{V}\ge c$ $$ {M}_p\frac{\mid \nabla V\mid }{V}\ge c $$ . Remarkably, the TCC bound $c\ge \frac{2}{\sqrt{\left(d-1\right)\left(d-2\right)}}$ $$ c\ge \frac{2}{\sqrt{\left(d-1\right)\left(d-2\right)}} $$ is then perfectly satisfied for d ≥ 4, with several saturation cases. However, we observe a violation of this bound in d = 3. We finally comment on related proposals in the literature, on the swampland distance conjecture and its decay rate, and on the so-called accelerated expansion bound.