*(English)*Zbl 0904.51002

A $t$-fold blocking set $B$ in a projective plane is a set of points such that each line contains at least $t$ points of $B$ and some line contains exactly $t$ points of $B$. A $(k,n)$-arc is a set of $k$ points such that some $n$, but no $n+1$ of them, are collinear. Define ${m}_{n}(2,q)$ to be the maximum size of a $(k,n)$-arc in $\text{PG}(2,q)$. Note that determining ${m}_{n}(2,q)$, or the minimum size of a $(q+1-n)$-fold blocking set, are equivalent problems. In the paper such a difficult question is dealt with. The value of ${m}_{n}(2,q)$ was already known for $2\le n<q\le 9$, and for $q>9$ only in few particular cases.

The main theorems of the paper are the following:

Theorem 1. Let $B$ be a $t$-fold blocking set in $\text{PG}(2,q)$. If $B$ contains no line then it has at least $tq+\sqrt{tq}+1$ points.

Theorem 2. Let $B$ be a $t$-fold blocking set in $\text{PG}(2,p)$ with $p>3$ prime. (i) If $t<p/2$ then $\left|B\right|\ge (t+\frac{1}{2})(p+1)$. (ii) If $t>p/2$ then $\left|B\right|\ge (t+1)p$.

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 ${m}_{n}(2,11)$ and ${m}_{n}(2,13)$ for some $n$ and bounds in other cases. A table of all known values of ${m}_{n}(2,q)$ for $2\le n<q\le 13$ containing complete references is given.

##### MSC:

51E21 | Blocking sets, ovals, $k$-arcs |