On a problem of Erdős and Lovász: Random lines in a projective plane


Letn(k) be the least size of an intersecting family ofk-sets with cover numberk, and let ℝ k denote any projective plane of orderk−1.


There is a constant A such that ifH is a random set ofmAklogk lines from ℝ k then PrH<)→0(k→∞).


If there exists a ℝ k thenn(k)=O(klogk). These statements were conjectured by P. Erdős and L. Lovász in 1973.

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Supported in part by NSF-DMS87-83558 and AFOSR grants 89-0066, 89-0512 and 90-0008

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Kahn, J. On a problem of Erdős and Lovász: Random lines in a projective plane. Combinatorica 12, 417–423 (1992). https://doi.org/10.1007/BF01305234

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