A completion problem for finite affine planes

Abstract

A partial affine plane (PAP) of ordern is ann 2-setS of points together with a collection ofn-subsets ofS called lines such that any two lines meet in at most one point. We obtain conditions under which a PAP with nearlyn 2+n lines can be completed to an affine plane by adding lines. In particular, we make use of Bruck’s completion condition for nets to show that certain PAP’s with at leastn 2+n−√n can be completed and that forn≠3 any PAP withn 2+n−2 lines can be completed.

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Dow, S. A completion problem for finite affine planes. Combinatorica 6, 321–325 (1986). https://doi.org/10.1007/BF02579258

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AMS subject classification (1980)

  • 05 B 25
  • 51 E 15