Suppose H is a linear 3-uniform hypergraph, i

Definitions and mathematical statement

A finite $3$-uniform hypergraph $H=(V,\mathcal E)$ has three vertices in each edge. It is linear if distinct edges share at most one vertex. Say that it has a faithful straight-triangle embedding in $\mathbb R^3$ if there is an injective placement $p:V\to\mathbb R^3$ such that every edge $e$ spans a nondegenerate closed triangle $T_e=\operatorname{conv}p(e)$ and \[ T_e\cap T_f=\operatorname{conv}p(e\cap f) \qquad(e\ne f), \] where the convex hull of the empty set is empty. Thus disjoint edges give disjoint triangles, and two incident triangles meet only at their common vertex. Also require $p(v)\notin T_e$ whenever $v\notin e$.

Let $h(n)$ be the maximum number of edges in an $n$-vertex linear $3$-uniform hypergraph admitting such an embedding. Kalai's question asks whether \[ h(n)=o(n^2). \] Precisely, for every $\varepsilon>0$, must there be $n_0$ such that every eligible hypergraph with $n\ge n_0$ has at most $\varepsilon n^2$ edges?

Linearity alone gives only a quadratic upper bound, by counting pairs of vertices. The question is whether the geometric realization forces a vanishing quadratic density. Intersections of triangle interiors, overlaps along unprescribed segments, and a vertex lying inside a nonincident triangle are excluded by faithfulness; allowing these would change the geometric restriction substantially.

Short English statement

Does faithful realization by triangles in three-dimensional space force every linear 3-uniform hypergraph to have o(n²) edges?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3100006 (AMR-030-0006), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Cooper - Combinatorial Problems I Like (2020 snapshot), Geometry, source-order bullet 6. Original source indexed by AMR-030-0006. https://people.math.sc.edu/cooper/combprob.html.