Question 7
Let $n\gt 1$ be an integer. Let $T$ be the set of all ordered triples $(x,y,z)$ of pairwise distinct positive integers with $1\le x,y,z\le2n$. A set $\mathcal A$ of ordered pairs $(u,v)$ is called linked to $T$ if, for every $(x,y,z)\in T$,
$$\{(x,y),(y,z),(z,x)\}\cap\mathcal A\ne\varnothing.$$
a) Find $|T|$.
b) Prove that there exists a set linked to $T$ with exactly $2n(n-1)$ elements.
c) Prove that every set linked to $T$ has at least $2n(n-1)$ elements.
$$\{(x,y),(y,z),(z,x)\}\cap\mathcal A\ne\varnothing.$$
a) Find $|T|$.
b) Prove that there exists a set linked to $T$ with exactly $2n(n-1)$ elements.
c) Prove that every set linked to $T$ has at least $2n(n-1)$ elements.