On Grids in Point-Line Arrangements in the Plane

Mirzaei, Mozhgan; Suk, Andrew

doi:10.1007/s00454-020-00231-x

Cited by 5 publications

(6 citation statements)

References 19 publications

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“…, l ′ k ∈ L such that p i,j ∈ l i ∩ l ′ j . In [23], it was shown that any set of n points and n lines in the plane that does not contain a k × k-grid has at most O(n 4/3−ε ) incidences, where ε depends on k. A straightforward adaptation of Lemma 9 gives the following. Theorem 13.…”

Section: Discussionmentioning

confidence: 94%

Hasse diagrams with large chromatic number

Suk¹,

Tomon²

2020

Preprint

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For every positive integer n, we construct a Hasse diagram with n vertices and chromatic number Ω(n 1/4 ), which significantly improves on the previously known best constructions of Hasse diagrams having chromatic number Θ(log n). In addition, if we also require that our Hasse diagram has girth at least k ≥ 5, we can achieve a chromatic number of at least n 1 2k−3 +o(1) . These results have the following surprising geometric consequence. They imply the existence of a family C of n curves in the plane such that the disjointness graph G of C is triangle-free (or have high girth), but the chromatic number of G is polynomial in n. Again, the previously known best construction, due to Pach, Tardos and Tóth, had only logarithmic chromatic number.

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Section: Discussionmentioning

confidence: 94%

Hasse diagrams with large chromatic number

Suk¹,

Tomon²

2020

Preprint

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“…As a corollary of Theorem 1.2 for k = 3, we get that there exists an arrangement of n points and n lines in the plane with no C 6 in the incidence graph while determining Ω(n 1+ 1 6 ) incidences. It is worth mentioning that one can follow the construction of Theorem 1.5 in [12] to get a construction of an arrangement of n points and n lines in the plane where their incidence graph is C 6 -free while it determines Ω(n 1+ 1 7 ) incidences.…”

Section: Discussionmentioning

confidence: 99%

Constructions of point-line arrangements in the plane with large girth

Mirzaei,

Suk,

Verstraëte

2019

Preprint

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A classical result by Erdős, and later on by Bondy and Simonivits, states that every n-vertex graph with no cycle of length 2k has at most O(n 1+1/k ) edges. This bound is known to be tight when k ∈ {2, 3, 5}, but it is a major open problem in extremal graph theory to decide if this bound is tight for all k.In this paper, we study the effect of forbidding short even cycles in incidence graphs of point-line arrangements in the plane. It is not known if the Erdős upper bound stated above can be improved to o(n 1+1/k ) in this geometric setting, and in this note, we establish non-trivial lower bounds for this problem by modifying known constructions arising in finite geometries. In particular, by modifying a construction due to Labeznik and Ustimenko, we construct an arrangement of n points and n lines in the plane, such that their incidence graph has girth at least k+5, and determines at least Ω(n 1+ 4 k 2 +6k−3 ) incidences. We also apply the same technique to Wenger graphs, which gives a better lower bound for k = 5.

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“…A recent result of Mirzaei and Suk [13] can be seen as quantitative variant of Theorem 1.2, for specific types of subgraphs.…”

Section: Introductionmentioning

confidence: 95%

A structural Szemerédi-Trotter Theorem for Cartesian Products

Sheffer¹,

Silier²

2021

Preprint

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We study configurations of n points and n lines that form Θ(n 4/3 ) incidences, when the point set is a Cartesian product. We prove structural properties of such configurations, such that there exist many families of parallel lines or many families of concurrent lines. We show that the line slopes have multiplicative structure or that many sets of y-intercepts have additive structure. We introduce the first infinite family of configurations with Θ(n 4/3 ) incidences. We also derive a new variant of a different structural point-line result of Elekes.Our techniques are based on the concept of line energy. Recently, Rudnev and Shkredov introduced this energy and showed how it is connected to point-line incidences. We also prove that their bound is tight up to sub-polynomial factors.

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On Grids in Point-Line Arrangements in the Plane

Cited by 5 publications

References 19 publications

Hasse diagrams with large chromatic number

Hasse diagrams with large chromatic number

Constructions of point-line arrangements in the plane with large girth

A structural Szemerédi-Trotter Theorem for Cartesian Products

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