# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Multiple blocking sets and arcs in finite planes. (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 $\left(k,n\right)$-arc is a set of $k$ points such that some $n$, but no $n+1$ of them, are collinear. Define ${m}_{n}\left(2,q\right)$ to be the maximum size of a $\left(k,n\right)$-arc in $\text{PG}\left(2,q\right)$. Note that determining ${m}_{n}\left(2,q\right)$, or the minimum size of a $\left(q+1-n\right)$-fold blocking set, are equivalent problems. In the paper such a difficult question is dealt with. The value of ${m}_{n}\left(2,q\right)$ was already known for $2\le n, 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 $\text{PG}\left(2,q\right)$. 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 $\text{PG}\left(2,p\right)$ with $p>3$ prime. (i) If $t then $|B|\ge \left(t+\frac{1}{2}\right)\left(p+1\right)$. (ii) If $t>p/2$ then $|B|\ge \left(t+1\right)p$.

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

##### MSC:
 5.1e+22 Blocking sets, ovals, $k$-arcs
##### Keywords:
multiple blocking sets; arcs; lacunary polynomials