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Summary: Economical coverings of sets of lattice points
Noga Alon
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
Abstract
Let t(n, d) be the minimum number t such that there are t of the nd
lattice points
{(x1, . . . , xd) : 1 xi n}
so that the t
2 lines that they determine cover all the above nd
lattice points. We prove that
for every integer d 2 there are two positive constants c1 = c1(d) and c2 = c2(d) such that for
every n
c1nd(d-1)/(2d-1)
t(n, d) c2nd(d-1)/(2d-1)
log n.
The special case d = 2 settles a problem of Erd¨os and Purdy.
Research supported in part by a United States Israel BSF Grant and by a Bergmann Memorial Grant
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