Maximum odd number of subsets, each intersects exactly half of the others

Not an answer, but too long for a comment.

First, here's a graph-theoretic interpretation. Define a graph with a node for each subset of $\{1,\dots,n\}$ and an edge for each pair of distinct nodes that have nonempty intersection. The problem is to find a largest subset $S$ of nodes such that both the subgraph induced by $S$ and its complement are $k$-regular.

Now here's an integer linear programming formulation. Let $N_i$ be the set of neighbors of node $i$, and let $\overline{k}=(2^n-1)/2$ be an upper bound on integer decision variable $k$. Let binary decision variable $x_i$ indicate whether node $i$ is selected. The problem is to maximize $k$ subject to \begin{align} \sum_i x_i &= 2k+1\\ -\overline{k}(1-x_i) \le \sum_{j \in N_i} x_j - k &\le |N_i|(1-x_i) &\text{for all $i$}\\ k &\in [0, \overline{k}] \cap \mathbb{Z} \end{align} The first constraint specifies that $2k+1$ nodes are selected. The second "big-M" constraint enforces the implication $x_i=1 \implies \sum_{j \in N_i} x_j = k$. That is, if node $i$ is selected then exactly $k$ of its neighbors are also selected. The third constraint enforces bounds and integrality of $k$. The optimal values reported above for $n\in\{1,\dots,9\}$ were obtained by solving this formulation.