For graphs with no induced T (forest with broom components or any forest) and no induced H (complete multipartite or bipartite), χ(G) is at most C times R(α(H), ω(G)+1) for a constant C depending only on T and H.
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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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Ramsey-type $\chi$-bounds for $\chi$-bounded graph classes
For graphs with no induced T (forest with broom components or any forest) and no induced H (complete multipartite or bipartite), χ(G) is at most C times R(α(H), ω(G)+1) for a constant C depending only on T and H.
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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.