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- W2106248614 abstract "For every $$k>3$$k>3, we give a construction of planar point sets with many collinear $$k$$k-tuples and no collinear $$(k+1)$$(k+1)-tuples. We show that there are $$n_0=n_0(k)$$n0=n0(k) and $$c=c(k)$$c=c(k) such that if $$nge n_0$$nźn0, then there exists a set of $$n$$n points in the plane that does not contain $$k+1$$k+1 points on a line, but it contains at least $$n^{2-({c}/{sqrt{log n}})}$$n2-(c/logn) collinear $$k$$k-tuples of points. Thus, we significantly improve the previously best known lower bound for the largest number of collinear $$k$$k-tuples in such a set, and get reasonably close to the trivial upper bound $$O(n^2)$$O(n2)." @default.
- W2106248614 created "2016-06-24" @default.
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- W2106248614 date "2013-07-10" @default.
- W2106248614 modified "2023-10-12" @default.
- W2106248614 title "Many Collinear $$k$$ k -Tuples with no $$k+1$$ k + 1 Collinear Points" @default.
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- W2106248614 doi "https://doi.org/10.1007/s00454-013-9526-9" @default.
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