On some points-and-lines problems and configurations

Abstract

Abstract. We apply an old method for constructing points-and-lines 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 tree-planting 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 56Introduction. Almost a century after Dudeney wrote these paragraphs, problems in incidence geometry continue to perplex both recreational and professional mathematicians, and the prospect of a uniform "direct and certain way of solving them" remains remote. Even for natural asymptotic questions, a wide gap often separates the best upper and lower bounds known. In this paper we construct some explicit point-and-line configurations that yield new lower bounds for two specific questions of this kind. Question 1, suggested by the recreational literature, asks: How many lines can meet n 2 points in the plane in at least n points each? Question 2 arises in the research literature [3]: If on each of N horizontal lines we choose (at most) N points, how many additional lines can contain N of these N 2 points? It turns out that an arrangement of 16 points in 15 lines of 4 (Figure 1 below), which has been known at least since 1908, naturally generalizes to configurations that not only give lower bounds for Question 1 but also improve on the previous records for Question 2. We also find a variation of this construction that yields a partial answer to Question 1 and a further improvement for the cases N = 12m = 12, 24, 36, . . . and N = 12m − 1 = 11, 23, 35, . . . of Question 2. By the construction in [3], the new results for Question 2 yield, for each N ≥ 5, improved lower bounds on the exponent in the asymptotic "orchard-planting" problem with N -point lines. Each of these arrangements exploits dihedral symmetry: the lines include all axes of symmetry, and every point lies on one of the axes and at least one pair of lines symmetrical with respect to this axis. This approach is at least a century old (we give specific citations later), but might still produce further new examples and results for modern incidence geometry.

Many Collinear $$k$$ k -Tuples with no $$k+1$$ k + 1 Collinear Points

“…In 1976 Grünbaum [13] 2) . For some 30 years this was the best bound when Ismailescu [14], Brass [2], and Elkies [5] consecutively improved Grünbaum's bound for k ≥ 5. However, similarly to Grünbaum's bound, the exponent was going to 1 as k went to infinity.…”

On Characterization of Graphs Structures Connected with Some Algebraic Properties

“…Petri nets make use of the qualities of directed bipartite graphs and other capabilities to allow mathematical demonstrations of system behaviour while also making simulations of the system simple to build [23]. According to the mathematical property that every two lines have at most one common point and every two points are collinear, Levi graphs need not have any cycles of length 4; thus, their girth has to be six or higher [24]. A bipartite graph's biadjacency matrix is a matrix with 1 for each pair of nodes that are contiguous and 0 for nonadjacent nodes.…”