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

Solymosi, József; Stojaković, Miloš

doi:10.1007/s00454-013-9526-9

Cited by 8 publications

(9 citation statements)

References 14 publications

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“…If the point set is already known to lie on a low-degree algebraic curve, then both conjectures hold [4,16]. On the other hand, Solymosi and Stojaković [19] proved that for any constant k, there can be Ω(n 2−ε ) lines with exactly k points, but no line with k + 1 points.…”

Section: Introductionmentioning

confidence: 99%

Bisector Energy and Few Distinct Distances

Lund

Sheffer

Zeeuw

2016

Discrete Comput Geom

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We introduce the bisector energy of an n-point set P in R 2 , defined as E(P) = (a, b, c, d) ∈ P 4 | a, b have the same perpendicular bisector as c, d .If no line or circle contains M (n) points of P, then we prove that for any ε > 0We also derive the lower bound E(P) = Ω(M (n)n 2 ), which matches our upper bound when M (n) is large. We use our upper bound on E(P) to obtain two rather different results:(i) If P determines O(n/ √ log n) distinct distances, then for any 0 < α ≤ 1/4, either there exists a line or circle that contains n α points of P, or there exist Ω(n 8/5−12α/5−ε ) distinct lines that contain Ω( √ log n) points of P. This result provides new information on a conjecture of Erdős [7] regarding the structure of point sets with few distinct distances.(ii) If no line or circle contains M (n) points of P, the number of distinct perpendicular bisectors determined by P is Ω min M (n) −2/5 n 8/5−ε , M (n) −1 n 2 . This appears to be the first higher-dimensional example in a framework for studying the expansion properties of polynomials and rational functions over R, initiated by Elekes and Rónyai [2].

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Section: Introductionmentioning

confidence: 99%

Bisector Energy and Few Distinct Distances

Lund

Sheffer

Zeeuw

2016

Discrete Comput Geom

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“…The following problem might be viewed as a first step towards understanding configurations having only triple points. Apparently, questions revolving around the same idea, have been present in combinatorics and discrete geometry for some time, see §1.1 in [18]. Problem 3.14.…”

Section: Furthermore If Tmentioning

confidence: 99%

Bounded Negativity and Arrangements of Lines

Bauer¹,

Rocco²,

Harbourne³

et al. 2014

Int Math Res Notices

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The Bounded Negativity Conjecture predicts that for any smooth complex surface X there exists a lower bound for the selfintersection of reduced divisors on X. This conjecture is open. It is also not known if the existence of such a lower bound is invariant in the birational equivalence class of X. In the present note we introduce certain constants H(X) which measure in effect the variance of the lower bounds in the birational equivalence class of X. We focus on rational surfaces and relate the value of H(P 2 ) to certain line arrangements. Our main result is Theorem 3.3 and the main open challenge is Problem 3.10. Problem 1.2 (Birational invariance of the BNC). Let X and Y be birationally equivalent projective surfaces. Does BNC hold for X if and only if it holds for Y ? In other words: is the bounded negativity property a birational invariant? Remark 1.3. Note, that a solution to the above problem is not known even if Y is the blow-up of X in a single point.Of course, if BNC is true in general, then the above problem has an affirmative solution. However, even in that situation, it is still of interest to know how the bounds b(X) and b(Y ) are related in terms of the complexity of a birational map between X and Y .

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“…The best construction for k = 3 come from irreducible cubic curves 4 . For higher k the best construction was given by Solymosi and Stojakovíc [14] and are projections of higher-dimensional subsets of the regular grid (selected, unlike ours, by taking concentric spheres). In the plane, our problem is dual to a colorful variant of Erdős's question.…”

Section: In Computer Visionmentioning

confidence: 99%

Consistent sets of lines with no colorful incidence

Bukh,

Goaoc,

Hubard

et al. 2018

Preprint

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

Cited by 8 publications

References 14 publications

Bisector Energy and Few Distinct Distances

Bisector Energy and Few Distinct Distances

Bounded Negativity and Arrangements of Lines

Consistent sets of lines with no colorful incidence

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