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%reference for an article :
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     p.\nobreak\ #7.\par}}
%reference for a book (livre) :
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\pageno=129

{
\eightpoint
\leftline{Publ. I.R.M.A. Strasbourg, $\oldstyle 1987$, 340/S--15}
\leftline{Actes $158^e$ S\'eminaire Lotharingien, p. 129--130}

}
\vskip 1.5cm
\centerline{{\bf ON A PROBLEM ABOUT COVERING}}
\vskip 5pt
\centerline{{\bf LINES BY SQUARES}}
\vskip 2.8mm
\centerline{\sevenrm BY}
\vskip 2.8mm
\centerline{W{\sevenrm ALTER} KERN and A{\sevenrm LFRED} WANKA}
\vskip 1cm minus .5cm
{\bf Abstract}. ---
Let $S$ be the square $[0,n]^2$ of side length
$n\in  {\bf  N}$ and let
${\cal S}=\{ S_1,\ldots,S_t\} $
be a set of unit squares lying inside ${S}$, whose sides
are parallel to those of
$S$. The set $\cal S$ is called a line cover, if
every line intersecting $S$ also intersetcs some
$S_i\in  {\cal S}$.
Let $\tau (n)$ denote the minimum cardinality of a line cover, and let
$\tau '(n)$ be defined in the same way, except that we restrict
our attention to lines which are parallel
to either one of the axes
or one of the diagonals of $S$. It has been
conjectured by L.F. T\'oth
that $\tau (n)=2n+0(1)$ and
I.~Bar\'anyi and
Z. F\"uredi that
$\tau (n)={3\over 2}n+0(1)$. We will prove instead,
$\tau '(n)={4\over 3}+0(1)$, and as
to T\'oth's conjecture, we will exhibit
a ``non integer" solution to a related
LP-relaxation, which has size equal to ${3\over 2}+0(1)$.
\vskip 1cm
{
\eightpoint
\centerline{REFERENCES}
\nobreak
\vskip .5cm
\article T|Fejes T\'oth (L.)|Remarks
on a dual of Tarski's plank problem|Mat. Lapok|25|1974|13--20|
\divers M|Moser (W.O.) and Pach (J.)|Research
Problems in Discrete Geometry, Problem 84, Montr\'eal,
$\oldstyle 1984$, mimeographed|
\divers BF|B\'ar\'anyi (L.) and F\"uredi (Z.)|Covering
all secants of a square, Rutcor Research Report 3--36,
Rutgers Univ., New Brunswick, N.J., March $\oldstyle 1986$|

}

\vskip 1cm
\nobreak
\rightline{\vbox{\halign{#\hfill\cr
                         Walter  Kern u. Alfred  Wanka,\cr
                         Mathematisches Institut,\cr
                         der Universit\"at zu K\"oln,\cr
                         Weyertal 86--90,\cr
                         D-5000 K\"oln.\cr}}\quad}

\bye

