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Update 3

Given this alternate way of thinking about pooling designs, the detection capacity of $G$ can be defined as the largest integer $c$ such that no edge $e\in E(G)$ is contained in the union of at most $c$ edges of $E(G)\setminus \{e\}$. So if we want a pooling design with testing capacity $c$ which uses $t$ tests, we want a hypergraph on $t$ vertices with as many edges as possible such that no edge $e\in E(G)$ is contained in the union of at most $c$ edges of $E(G)\setminus \{e\}$. It turns out that this problem was studied in Erdős, Paul; Frankl, P.; Füredi, Z., Families of finite sets in which no set is covered by the union of (r) others, Isr. J. Math. 51, 79-89 (1985). ZBL0587.05021.

For example, let's compare the $K_8$-design to the hypercube design with $D=3$. In the $K_8$-design, each edge is a sample (there are 28), each vertex is a test pooling the samples which are incident with that vertex (there are 8), each test pools 7 samples (since the degree of each vertex is 7), and each sample will be used twice (since $K_8$ is 2-uniform). As I mentioned in a comment, this is better than the hypercube design in every parameter. Also you can see that if exactly one sample is infected, say the edge $\{i,j\}$, then exactly two tests (test $i$ and test $j$) will come back positive.

For example, let's compare the $K_8$-design to the hypercube design with $D=3$. In the $K_8$-design, each edge is a sample (there are 28), each vertex is a test pooling the samples which are incident with that vertex (there are 8), each test pools 7 samples (since the degree of each vertex is 7), and each sample will be used twice (since $K_8$ is 2-uniform). As I mentioned in a comment, this is better than the $D=3$ hypercube design in every parameter. Also you can see that if exactly one sample is infected, say the edge $\{i,j\}$, then exactly two tests (test $i$ and test $j$) will come back positive.

Update. (It seems best to update my previous answer rather than write a new one.)

In particular, this proves that every pooling design on $4\leq n\leq 6$ vertices has at least 4 edges, every pooling design on $7\leq n\leq 10$ vertices has at least 5 edges, etc.

Update 1. (It seems best to update my previous answer rather than write a new one.)

In particular, this proves that every pooling design on $4\leq n\leq 6$ vertices has at least 4 edges, every pooling design on $7\leq n\leq 10$ vertices has at least 5 edges, etc.

Update 2.

Again, after considering some more, I now think it's clearer to just remain in the setting of the hypergraph $G$ and forget about taking the dual.

For example, let's compare the $K_8$-design to the hypercube design with $D=3$. In the $K_8$-design, each edge is a sample (there are 28), each vertex is a test pooling the samples which are incident with that vertex (there are 8), each test pools 7 samples (since the degree of each vertex is 7), and each sample will be used twice (since $K_8$ is 2-uniform). As I mentioned in a comment, this is better than the hypercube design in every parameter. Also you can see that if exactly one sample is infected, say the edge $\{i,j\}$, then exactly two tests (test $i$ and test $j$) will come back positive.

For another example, let's compare the $K_{13}$-design to the hypercube design with $D=4$. The $D=4$ hypercube design handles 81 samples using 12 tests each of which has size 27 and each sample is used 4 times. The $K_{13}$-design handles 78 samples using 13 tests, but each test has size 12 and each sample is only used 2 times.

For a final example, let's compare the $K_{9,9}$-design (that is, a complete bipartite graph with 9 vertices in each part) to the $D=4$ hypercube design. The $K_{9,9}$-design handles 81 samples using 18 tests, each of which has size 9 and each sample is used 2 times; however, this design has the additional feature that if three tests come back positive, then we will know exactly which two samples are infected. Neither the $K_{13}$-design, nor the $D=4$ hypercube design have that property.