Given a natural number n>1. I'd like to find a set $\phi = \{s_1,s_2, \cdots , s_m \} \subset \{1,2,\cdots , n \} $ with $m > 1$ that minimizes the following quantity: $$ S_{\phi} = \frac{\max_{k\in\mathbb{N}^*}\frac{\cdot\operatorname{Card}\left[\{s_1 + k, s_2+k, \cdots , s_m+k\} \cap \phi \right]}{m}}{\frac{m}{n}} = \max_{k\in\mathbb{N}^*} \frac{n \cdot\operatorname{Card}\left[\{s_1 + k, s_2+k, \cdots , s_m+k\} \cap \phi \right]}{m^2}$$
Anyone has an idea how to treat this kind of problems?
It looks like you are searching for minimally symmetric subset of a discrete segment with the group of symmetries consisting of the translations. Such problems and approaches to solve them are known (telling the truth, there were dealt by my professor :-) ), see, for instance, the references ([BVV, §1-2, 5], [BP, Sect. 8], and [B, Sect. 6]).
References
[B] T.Banakh, Symmetry and Colorings: Some Results and Open Problems, II.
[BVV] T.Banakh, O.Verbitski, Ya.Vorobets, Ramsey treatment of symmetry // Electronic J. of Combinatorics. 7:1 (2000) R52. – 25p.
[BP] T.Banakh, I.Protasov. Symmetry and colorings: some results and open problems // Izv. Gomel Univ. Voprosy Algebry. 2001. Issue 4(17). P.5–16.