Abstract
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.
Theorem
There is a constant A such that ifH is a random set ofm ≥Aklogk lines from ℝ k then Pr(τH<)→0(k→∞).
Corollary
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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References
P. Erdős: On the combinatorial problems I would most like to see solved,Combinatorica 1 (1981), 25–42.
P. Erdős, andL. Lovász: Problems and results on 3-chromatic hypergraphs and some related questions, in:Infinite and Finite Sets (Proc. Colloq. Math. Soc. J. Bolyai 10, Keszthely, Hungary, 1973), A. Hajnal et. al. (eds.), North Holland, Amsterdam, 1975, 609–627.
Z. Füredi: Matching and covers in hypergraphs,Graphs and Combinatorics 4 (1988), 115–206.
J. Kahn: On a theorem of Frankl and Rödl, in preparation.
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Supported in part by NSF-DMS87-83558 and AFOSR grants 89-0066, 89-0512 and 90-0008