Existence of an optimal data placement for arbitrary parameters

Let $V$ be a set of $v$ servers, and let a data placement be a family of $b$ blocks, each of size $k$, on $V$. For a placement $(V,\mathcal{F})$, let

$$P(\mathcal{F},x)=\sum_{j=0}^{k}v_jx^j,$$

where $v_j$ is the number of ordered pairs of blocks intersecting in exactly $j$ elements and $x\geq1$. Optimal data placement conjecture. For any nonnegative integers $v$, $k$ and $b$, there exists an optimal data placement consisting of $b$ blocks of size $k$ on a $v$-set, meaning one that minimizes $P(\mathcal{F},x)$ for every $x\geq1$. The conjecture asserts simultaneous optimality for all failure probabilities; the paper studies this problem and, for triple replication, establishes existence in broad parameter ranges while leaving possible exceptional cases.