Economical coverings of sets of lattice points
نویسنده
چکیده
Let t(n, d) be the minimum number t such that there are t of the n lattice points {(x1, . . . , xd) : 1 ≤ xi ≤ n} so that the ( t 2 ) lines that they determine cover all the above n 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 c1n d(d−1)/(2d−1) ≤ t(n, d) ≤ c2n log n. The special case d = 2 settles a problem of Erdös and Purdy. ∗Research supported in part by a United States Israel BSF Grant and by a Bergmann Memorial Grant
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