What is the difference between saying there is no ϵ>0 such that a problem can be solved in $O(n^{2-\epsilon})$ time and $n^{2-o(1)}$ or $\Omega(n^2)$?
It is conjectured that 3SUM cannot be solved in time $O(n^{2-\epsilon})$ for any $\epsilon &gt; 0$; equivalently, it requires time $n^{2-o(1)}$. This is not the same as the stronger conjecture that 3SUM requires time $\Omega(n^2)$. Indeed, the latter conjecture (which was the original form of the 3SUM conjecture) is false: Grønlund and Pettie came up with an $O(n^2/(\log n/\log\log n)^{3/2})$ algorithm (there were improvements since).
The statement &quot;problem X requires time $n^{2-o(1)}$&quot; states that there exists a function $f(n) = o(1)$ such that any algorithm for X runs in time at least $n^{2-f(n)}$. In particular, there is no $O(n^{2-\epsilon})$ algorithm, for any $\epsilon &gt; 0$. The converse should also hold, I believe – could be a  fun exercise to show.