General Position for Planar Line Arrangements and $HD_2(p,3)$
Prior state unknown→proved
improved bounds; the exact Hadwiger-Debrunner numbers remain open
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geometry-topology / Discrete geometry
General Position for Planar Line Arrangements and $HD_2(p,3)$
For every $\delta > 0$ and infinitely many $n$ there is a set of $n$ lines in the plane with no intersecting quadruple such that every subset of size at least $n^{4/5+\delta}$ contains three concurrent lines. This improves the bound for a dual form of a theorem of Balogh and Solymosi, and yields an improved lower bound for the Hadwiger-Debrunner number $HD_2(p,3)$.
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For every $\delta > 0$ and infinitely many $n$ there is a set of $n$ lines in the plane with no intersecting quadruple such that every subset of size at least $n^{4/5+\delta}$ contains three concurrent lines. This improves the bound for a dual form of a theorem of Balogh and Solymosi, and yields an improved lower bound for the Hadwiger-Debrunner number $HD_2(p,3)$.
improved bounds; the exact Hadwiger-Debrunner numbers remain open
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