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

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 ofmAklogk lines from ℝ k then PrH<)→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.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    P. Erdős: On the combinatorial problems I would most like to see solved,Combinatorica 1 (1981), 25–42.

    Google Scholar 

  2. [2]

    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.

    Google Scholar 

  3. [3]

    Z. Füredi: Matching and covers in hypergraphs,Graphs and Combinatorics 4 (1988), 115–206.

    Google Scholar 

  4. [4]

    J. Kahn: On a theorem of Frankl and Rödl, in preparation.

Download references

Author information

Affiliations

Authors

Additional information

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

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

AMS subject classification code 1991

  • 05 B 40
  • 05 C 65
  • 05 D 05
  • 51 E 15