It is possible to achieve $N+1$ days if and only if there is a [finite projective plane][1] with $N^2+N+1$ points.

<b>Given the plane:</b> Choose a special line $L_{\infty}$ in the plane. Let the $N+1$ points on $L_{\infty}$ be $(D_1, D_2, \dots, D_{N+1})$; these will index the days of the conference. Let the $N^2$ points not on $L_{\infty}$ be $(P_1, P_2, \ldots, P_{N^2})$, these will index the people. For each $D_k$, there are $N$ lines through $D_k$ other then $L_{\infty}$; each of these lines contains $N$ of the $P_i$'s. Use this seating arrangement on day $k$.
Since every pair $(P_i, P_j)$ is on a unique line, and that unique line intersect $L_{\infty}$ at a unique $D_k$, there is a unique day on which $i$ and $j$ sit together.

This construction is easily reversible: <b>Given the seating arrangement</b>, let the points of our underlying plane be $N^2$ points $P_i$ for the people and $N+1$ points $D_k$ for the days. Define a line to consist of $D_k$, together with all of the $P_i$ for people $i$ who sat together on day $k$. Also, add one more line $\{ D_1, D_2, \ldots, D_{N+1} \}$. It is straightforward to check that this obeys the axioms of a projective plane.

As the Wikipedia link above says, projective planes exist whenever $N$ is a prime power, and have been proved not to exist for $N=6$, $10$. The particular case of $N=6$, which you raise, is impossible. Moral: Conferences should be $5$ days long, not $7$ !


  [1]: https://en.wikipedia.org/wiki/Projective_plane#Finite_projective_planes