A -fold blocking set in a projective plane is a set of points such that each line contains at least points of and some line contains exactly points of . A -arc is a set of points such that some , but no of them, are collinear. Define to be the maximum size of a -arc in . Note that determining , or the minimum size of a -fold blocking set, are equivalent problems. In the paper such a difficult question is dealt with. The value of was already known for , and for only in few particular cases.
The main theorems of the paper are the following:
Theorem 1. Let be a -fold blocking set in . If contains no line then it has at least points.
Theorem 2. Let be a -fold blocking set in with prime. (i) If then . (ii) If then .
The latter theorem is a generalization of a result by A. Blokhuis [Bolyai Soc. Math. Stud. 2, 133-155 (1996; Zbl 0849.51005)] and uses the theory of lacunary polynomials. In some cases the bounds in Theorem 2 are sharp.
The author finds examples and proves further results that with the help of Theorem 2 yield the exact value of and for some and bounds in other cases. A table of all known values of for containing complete references is given.