Tripod packing

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $p=(x,y,z)\in\mathbb R^3$, define the tripod \[T(p)=\{(x+t,y,z):t\geq0\}\cup\{(x,y+t,z):t\geq0\}\cup\{(x,y,z+t):t\geq0\}.\] Thus its three arms are infinite closed rays in the positive coordinate directions. For an integer $n\geq1$, the allowed apices are the lattice points $\{1,\ldots,n\}^3$. Define $F(n)$ to be the maximum cardinality of a subset $A$ of this lattice cube such that $T(p)\cap T(q)=\varnothing$ for all distinct $p,q\in A$. The rays need not stay in the cube.

Statement recorded in the supplied source.

Determine $F(n)$, or its asymptotic growth as $n\to\infty$: how many pairwise disjoint coordinate tripods can have their apices in the $n\times n\times n$ lattice cube? [S2]

Short English statement

How many disjoint tripods with three infinite coordinate rays can have their apices among the lattice points of a cube?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200058 (AMR-071-0058). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Boris Aronov, Vida Dujmović, Pat Morin, Aurélien Ooms and Luís Fernando Schultz Xavier da Silveira, More Turán-Type Theorems for Triangles in Convex Point Sets (2018 version). Section 4.3, pp. 20–22. https://arxiv.org/pdf/1706.10193v3.