The Zarankiewicz number for these box-intersection hypergraphs is either Θ_r(t n^{r-1}) or Ω(t n^{r-1} log n / log log n) according to whether the direction families (F1,...,Fr) are 2-coherent.
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A dichotomy for hypergraph Zarankiewicz problems on axis-parallel boxes
The Zarankiewicz number for these box-intersection hypergraphs is either Θ_r(t n^{r-1}) or Ω(t n^{r-1} log n / log log n) according to whether the direction families (F1,...,Fr) are 2-coherent.