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co.combinatorics – What are environment friendly pooling designs for RT-PCR assessments?

This is not a complete respond, however too lengthy for a observation. I suppose it comes closest to attempting to respond Question 3 or the common query of whether or not the hypercube project can breathe improved.

**Definition** Given a hypergraph $G=({v_1, dots, v_n}, E)$, the *twin* of $G$ is the hypergraph $H$ with $V(H)=E(G)$ and $E(H)={{ein E(G): v_iin e}: iin [k]}$ (in different phrases, every verge of $H$ is a maximal assortment of edges from $G$ that are incident with a sole vertex).

Let $H_{n,ok}$ breathe the twin of $K_n^{ok}$, the whole $ok$-regular hypergraph on $n$ vertices. Note that the twin of $H_{n,ok}$ is isomorphic to $K_n^ok$.

(It appears to me that this hypergraph will need to have been studied earlier than, however I could not discover any reference to it. One workable lead is that $H_{4,2}$ is what you convene the *full quadrilateral*.)

**Claim 1.**

$H_{n,ok}$ is a $binom{n-1}{k-1}$-uniform $ok$-regular hypergraph with $binom{n}{ok}$ vertices and $n$ edges.

*Proof.* In $K_n^ok$, each vertex is incident with $binom{n-1}{k-1}$ edges, each verge has organize $ok$, there are $binom{n}{ok}$ edges, and $n$ vertices.$sq.$

**Claim 2.** $H_{n,ok}$ is a pooling project.

*Proof.*

Every vertex in $H_{n,ok}$ is incident with $ok$ edges, so $|x^*|=ok$. If $X$ is a clique of vertices with $|X|>1$ (which corresponds to a clique of a couple of verge in $K_n^ok$, which spans greater than $ok$ vertices in $K_n^ok$) then $|X^*|>ok$. So $x^*neq X^*$ if $|X|>1$.$sq.$

The compression price of $H_{n,ok}$ is $frac{n}{binom{n}{ok}}$ which is minimized when $ok=lfloor{n/2}rfloor$. Also point to that ratio of the uniformity to the variety of vertices is $binom{n-1}{k-1}/binom{n}{ok}=ok/n$. So there’s a tradeoff

when minimizing the compression price, because the uniformity and diploma expand after we expand $ok$.

Some extra examples: $H_{5,2}$ is 4-uniform with 10 vertices and 5 edges giving a compression ratio of $1/2$. $H_{6,3}$ is 10-uniform with 20 vertices and 6 edges, giving a compression ratio of $3/10$. $H_{7,3}$ is 15-uniform with 35 vertices and seven edges, giving a compression ratio of $1/5$. Note that the hypercube project with $D=3$ is 9-regular with 27 vertices and 9 edges and thus a compression ratio of 1/3, so $H_{6,3}$ and $H_{7,3}$ examine favorably on this illustration.

**Update 1**. (It appears greatest to replace my earlier respond reasonably than write a fresh one.)

After considering it over some extra, I cerebrate I’ve an alternate characterization of pooling designs which each makes it simpler to bridle if $H$ is a pooling project and elucidates some options of pooling designs. In specific, this provides a unostentatious proof of the Propositions in your respond.

**Claim 3** $H$ is a pooling project if and provided that $x^*notsubseteq y^*$ for all discrete $x,yin V(H)$.

*Proof.*

($Rightarrow$) Suppose there exists discrete $x,yin V(H)$ such that $x^*subseteq y^*$. Then $y^*={x,y}^*$ and thus $H$ shouldn’t be a pooling project.

($Leftarrow$) Suppose $H$ shouldn’t be a pooling project; that’s, suppose there exists $yin V(H)$ and $Ysubseteq V(H)$ with $Yneq {y}$ such that $y^*=Y^*$. Since $Yneq {y}$, there exists $xin Y$ such that $xneq y$. Since $xin Y$, we now have $x^*subseteq Y^*=y^*$.

$sq.$

**Corollary 1**

Let $H$ breathe a hypergraph and let $G$ breathe the twin of $H$.

$H$ is a pooling project if and provided that $enotsubseteq f$ for all discrete $e,fin E(G)$.

*Proof.*

($Rightarrow$) Suppose $H$ is a pooling project. Choose discrete $e,fin E(G)$ which coincide to discrete $x, yin V(H)$ respectively. Since $x^*notsubseteq y^*$, we now have $enotsubseteq f$.

($Leftarrow$) Suppose $enotsubseteq f$ for all discrete $e,fin E(G)$. Choose discrete $x,yin V(H)$ which coincide to discrete $e,fin E(G)$. Since $enotsubseteq f$, we now have $x^*notsubseteq y^*$.

$sq.$

**Corollary 2**

Let $H$ breathe a hypergraph with $e$ edges and $n$ vertices such that $binom{e}{lfloor{e/2}rfloor}<n$. Then $H$ shouldn’t be a pooling project.

*Proof.*

Let $G$ breathe the twin of $H$ and point to that $G$ has $e$ vertices and $n$ edges. Since $|E(G)|=n>binom{e}{lfloor{e/2}rfloor}=binom{lfloor/2rfloor}$, Sperner’s theorem implies that there exists discrete $e,fin E(G)$ such that $esubseteq f$. Thus $H$ shouldn’t be a pooling project by Corollary 1.

$sq.$

In specific, this proves that each pooling project on $4leq nleq 6$ vertices has no less than 4 edges, each pooling project on $7leq nleq 10$ vertices has no less than 5 edges, and many others.

**Update 2**.

Again, after contemplating some extra, I now cerebrate it is clearer to simply stay within the setting of the hypergraph $G$ and neglect about taking the twin.

For instance, let’s examine the $K_8$-design to the hypercube project with $D=3$. In the $K_8$-design, every verge is a specimen (there are 28), every vertex is a take a look at pooling the samples that are incident with that vertex (there are 8), every take a look at swimming pools 7 samples (because the diploma of every vertex is 7), and every specimen will breathe used twice (since $K_8$ is 2-uniform). As I discussed in a observation, that is higher than the $D=3$ hypercube project in each parameter. Also you may behold that if precisely one specimen is contaminated, say the verge ${i,j}$, then precisely two assessments (take a look at $i$ and take a look at $j$) will come advocate constructive.

For one other instance, let’s examine the $K_{13}$-design to the hypercube project with $D=4$. The $D=4$ hypercube project handles 81 samples utilizing 12 assessments every of which has measurement 27 and every specimen is used 4 occasions. The $K_{13}$-design handles 78 samples utilizing 13 assessments, however every take a look at has measurement 12 and every specimen is barely used 2 occasions.

For a last instance, let’s examine the $K_{9,9}$-design (that’s, a whole bipartite graph with 9 vertices in every sever) to the $D=4$ hypercube project. The $K_{9,9}$-design handles 81 samples utilizing 18 assessments, every of which has measurement 9 and every specimen is used 2 occasions; nonetheless, this project has the extra characteristic that if three assessments come advocate constructive, then we are going to know precisely which two samples are contaminated. Neither the $K_{13}$-design, nor the $D=4$ hypercube project have that property.

**Update 3**

Given this alternate route of occupied with pooling designs, the detection capability of $G$ can breathe outlined as the most important integer $c$ such that no verge $ein E(G)$ is contained within the union of at most $c$ edges of $E(G)setminus {e}$. So if we would like a pooling project with testing capability $c$ which makes use of $t$ assessments, we would like a hypergraph on $t$ vertices with as many edges as workable such that no verge $ein E(G)$ is contained within the union of at most $c$ edges of $E(G)setminus {e}$. It seems that this drawback was studied in *Erdős, Paul; Frankl, P.; Füredi, Z.*, **Families of finite units wherein no clique is roofed by the union of (r) others**, Isr. J. Math. 51, 79-89 (1985). ZBL0587.05021.

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