What is the maximum number of tripods that can be packed in a $n\times n\times n$ cube? The current best upper bound is $n^2/e^{\Omega(\log^\star n)}$. The current best lower-bound is $\Omega(n^{1.546})$. See Section 4.3 of this paper.

An Approach Using $\epsilon$-Nets

For any $a,b,c\in\N$, a $a \times b\times c$ a tripod packing $T$ is a subset of $[a]\times[b]\times[c]$ with the property that, for any pair $x,y\in T$, $x_i > y_i$ for at least two values of $i\in\lbrace 1,2,3\rbrace$ or $x_i < y_i$ for at least two values of $i\in\lbrace 1,2,3\rbrace$.

For an $n\times n\times n$ tripod packing $T$ and any $z\in[n]$, let $T_z=\lbrace (x,y):(z,y,z)\in T\rbrace$ and observe that $T_1,\ldots,T_n$ is a partition of $T$. Let $C_z(T)=\bigcup_{(x,y)\in T_z} \lbrace x,y\rbrace$

We say that $N\subset\lbrace 1,\ldots,n\rbrace$ is an $\epsilon$-net for $T$ if, for every $z\in \lbrack n\rbrack$, such that $|T_z|\ge\epsilon n$, $C_z(T)\cap N\neq\emptyset$.

If this were true, then we could use it to prove an upper bound on the size of $n\times n\times n$ tripod packings, as follows. Suppose $T$ is an $n\times n\times n$ tripod packing of size at least than $2\epsilon n^2$. For every $z$ such that $|T_z| \le \epsilon n$, remove $T_z$ from $T$ to obtain a tripod packing $T’$ of size at least $\epsilon n^2$ in which $|T’_z| =0$ or $|T’_z|\ge \epsilon n$ for every $z\in[n]$.

Let $N$ be an $\epsilon$-net for $T’$ and consider the subgrid $N