# How to prove that a problem can't be solved in time $\mathcal O(n^{1/2 - \epsilon})$?

I have seen many problems in graph theory and in other related fields which admit a sublinear (in input size) running time $\mathcal O(n^{1/2})$ algorithm, where $n$ is the input size. I am not sure but to me it seems that fine grained complexity may be one possibility to prove that there is no algorithm with running time $\mathcal O(n^{1/2 - \epsilon})$, where $\epsilon$ is greater than 0. The model of computation is RAM.

Question : How to prove that a problem can't be solved in running time $\mathcal O(n^{1/2 - \epsilon})$?

One possibility to get conditional hardness is to give a (fine-grained) reduction from a problem that is known to be hard in some sense. The result will then something along the lines of "if problem X admits an algorithm running in time $O(n^{1/2-\epsilon})$, then problem Y can be solved within some specific time bound". The "specific time bound" should be something we don't know how or believe that can be achieved.