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On empty convex polygons in a planar point set
 J. Comb. Theory, Ser. A
"... Let P be a set of n points in general position in the plane. Let Xk(P) denote the number of empty convex kgons determined by P. We derive, using elementary proof techniques, several equalities and inequalities involving the quantities Xk(P) and several related quantities. Most of these equalities a ..."
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Let P be a set of n points in general position in the plane. Let Xk(P) denote the number of empty convex kgons determined by P. We derive, using elementary proof techniques, several equalities and inequalities involving the quantities Xk(P) and several related quantities. Most of these equalities and inequalities are new, except for a few that have been proved earlier using a considerably more complex machinery from matroid and polytope theory, and algebraic topology. Some of these relationships are also extended to higher dimensions. We present several implications of these relationships, and discuss their connection with several longstanding open problems, the most notorious of which is the existence of an empty convex hexagon in any point set with sufficiently many points. 1
On some pointsandlines problems and configurations
, 2006
"... Abstract. We apply an old method for constructing pointsandlines configurations in the plane to study some recent questions in incidence geometry. What are known as “Points and Lines ” puzzles are found very interesting by many people. The most familiar example, here given, to plant nine trees so ..."
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Abstract. We apply an old method for constructing pointsandlines configurations in the plane to study some recent questions in incidence geometry. What are known as “Points and Lines ” puzzles are found very interesting by many people. The most familiar example, here given, to plant nine trees so that they shall form ten straight rows with three trees in every row, is attributed to Sir Isaac Newton, but the earliest collection of such puzzles is, I believe, in a rare little book that I possess — published in 1821 — Rational Amusement for Winter Evenings, by John Jackson. The author gives ten examples of “Trees planted in Rows.” These treeplanting puzzles have always been a matter of great perplexity. They are real “puzzles, ” in the truest sense of the word, because nobody has yet succeeded in finding a direct and certain way of solving them. They demand the exercise of sagacity, ingenuity, and patience, and what we call “luck ” is also sometimes of service. — H.E. Dudeney, Amusements in Mathematics (1917) [8], page 56 Introduction. Almost a century after Dudeney wrote these paragraphs, problems in incidence geometry continue to perplex both recreational and professional mathematicians, and the
Many collinear ktuples with no k + 1 collinear points
, 2013
"... For every k> 3, we give a construction of planar point sets with many collinear ktuples and no collinear (k + 1)tuples. We show that there are n0 = n0(k) and c = c(k) such that if n ≥ n0, then there exists a set of n points in the plane that does not contain k + 1 points on a c log n line, but it ..."
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For every k> 3, we give a construction of planar point sets with many collinear ktuples and no collinear (k + 1)tuples. We show that there are n0 = n0(k) and c = c(k) such that if n ≥ n0, then there exists a set of n points in the plane that does not contain k + 1 points on a c log n line, but it contains at least n 2− collinear ktuples of points. Thus, we significantly improve the previously best known lower bound for the largest number of collinear ktuples in such a set, and get reasonably close to the trivial upper bound O(n2). 1