Noncollinear Set Research Articles

Chen-Chvátal’s conjecture for graphs with restricted girth

A classic result in Euclidean geometry asserts that every non-collinear set of n points in the Euclidean plane determines at least n distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with an appropriate definition of line. This conjecture remains open even for graph metrics. In this paper, sufficient conditions on the girth to guarantee that a graph satisfies the conjecture are stated. We first study the existence of a universal line generated by an edge. Then, we focus on graphs of order n and maximum girth g with respect to their radius r. We prove that graphs with minimum degree δ ≥ 4 and girth g = 2r+1 or g = 2r have at least n distinct lines and we also partially solve the case when δ = 3. Graphs with cut-vertices are also studied, providing several upper bounds on the diameter, in terms of the girth, in order to assure that the graph satisfies the Chen-Chvátal conjecture. Finally, we prove that any triangle-free graph of order n, diameter 3 and minimum degree δ ≥ 4 has more than n distinct lines and we also partially solve the case when δ = 3.

A class of self-affine tiles in [formula omitted] that are tame balls revisited

The author of this paper and coauthors in 2022 studied a family of self-affine tiles in Rd with noncollinear digit sets, and gave a sufficient and necessary condition for such tiles to be tame balls. We in this paper mainly present a simpler proof of such equivalent condition. We replace quadric surfaces by some zigzag planes, and redefine the quasi-invariant plane which plays a key role in the construction of the desired homeomorphism. This adjustment greatly simplifies the proof.

A class of self-affine tiles in [formula omitted] that are d-dimensional tame balls

We study a family of self-affine tiles in Rd (d≥2) with noncollinear digit sets, which naturally generalizes a two-dimensional class studied originally by Deng and Lau and its extension to R3 by the authors. By using Brouwer's invariance of domain theorem, along with a tool which we call horizontal distance, we obtain necessary and sufficient conditions for the tiles to be d-dimensional tame balls. This answers positively the conjecture in an earlier paper by the authors stating that a member in a certain class of self-affine tiles is homeomorphic to a d-dimensional ball if and only if its interior is connected.

A Sylvester–Gallai Result for Concurrent Lines in the Complex Plane

We show that if a finite non-collinear set of points in \(\mathbb {C}^2\) lies on a family of m concurrent lines, and if one of those lines contains more than \(m-2\) points, there exists a line passing through exactly two points of the set. The bound \(m-2\) in our result is optimal. Our main theorem resolves a conjecture of Frank de Zeeuw, and generalizes a result of Kelly and Nwankpa.

Topological Properties of a Class of Higher-dimensional Self-affine Tiles

AbstractWe construct a family of self-affine tiles in $\mathbb{R}^{d}$ ($d\geqslant 2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Q.-R. Deng and K.-S. Lau in $\mathbb{R}^{2}$, and its extension to $\mathbb{R}^{3}$ by the authors. We obtain necessary and sufficient conditions for the tiles to be connected and for their interiors to be contractible.

NEW RESULTS ON SUM‐PRODUCT TYPE GROWTH OVER FIELDS

We prove a range of new sum-product type growth estimates over a general field $\mathbb{F}$, in particular the special case $\mathbb{F}=\mathbb{F}_p$. They are unified by the theme of "breaking the $3/2$ threshold", epitomising the previous state of the art. These estimates stem from specially suited applications of incidence bounds over $\mathbb{F}$, which apply to higher moments of representation functions. We establish the estimate $|R[A]| \gtrsim |A|^{8/5}$ for cardinality of the set $R[A]$ of distinct cross-ratios defined by triples of elements of a (sufficiently small if $\mathbb{F}$ has positive characteristic, similarly for the rest of the estimates) set $A\subset \mathbb{F}$, pinned at infinity. The cross-ratio naturally arises in various sum-product type questions of projective nature and is the unifying concept underlying most of our results. It enables one to take advantage of its symmetry properties as an onset of growth of, for instance, products of difference sets. The geometric nature of the cross-ratio enables us to break the version of the above threshold for the minimum number of distinct triangle areas $Ouu'$, defined by points $u,u'$ of a non-collinear point set $P\subset \mathbb{F}^2$. Another instance of breaking the threshold is showing that if $A$ is sufficiently small and has additive doubling constant $M$, then $|AA|\gtrsim M^{-2}|A|^{14/9}$. This result has a second moment version, which allows for new upper bounds for the number of collinear point triples in the set $A\times A\subset \mathbb{F}^2$, the quantity often arising in applications of geometric incidence estimates.

De Bruijn–Erdős-type theorems for graphs and posets

A classical theorem of De Bruijn and Erdős asserts that any noncollinear set of n points in the plane determines at least n distinct lines. We prove that an analogue of this theorem holds for posets, where lines are defined using the natural betweenness relation in posets. More precisely, we obtain a bound on the number of lines depending on the height of the poset. The extremal configurations are also determined. Finally, we introduce a new notion of lines in graphs and show that our result for posets can be extended to this setting.

Lattice Embeddings of Planar Point Sets

Let $$\mathcal {M}$$M be a finite non-collinear set of points in the Euclidean plane, with the squared distance between each pair of points integral. Considering the points as lying in the complex plane, there is at most one positive square-free integer D, called the "characteristic" of $$\mathcal {M}$$M, such that a congruent copy of $$\mathcal {M}$$M embeds in $$\mathbb {Q}(\sqrt{-D})$$Q(-D). We generalize the work of Yiu and Fricke on embedding point sets in $$\mathbb {Z}^2$$Z2 by providing conditions that characterize when $$\mathcal {M}$$M embeds in the lattice corresponding to $$\mathcal {O}_{-D}$$O-D, the ring of integers in $$\mathbb {Q}(\sqrt{-D})$$Q(-D). In particular, we show that if the square of every ideal in $$\mathcal {O}_{-D}$$O-D is principal and the distance between at least one pair of points in $$\mathcal {M}$$M is integral, then $$\mathcal {M}$$M embeds in $$\mathcal {O}_{-D}$$O-D. Moreover, if $$\mathcal {M}$$M is primitive, so that the squared distances between pairs of points are relatively prime, and $$\mathcal {O}_{-D}$$O-D is a principal ideal domain, then $$\mathcal {M}$$M embeds in $$\mathcal {O}_{-D}$$O-D.

Lines, Betweenness and Metric Spaces

A classic theorem of Euclidean geometry asserts that any noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with a certain natural definition of lines in a metric space. We prove that in any metric space with n points, either there is a line containing all the points or there are at least \(\Omega (\sqrt{n})\) lines. This is the first lower bound on the number of lines in general finite metric spaces that grows faster than logarithmically in the number of points. In the more general setting of pseudometric betweenness, we prove a corresponding bound of \(\Omega (n^{2/5})\) lines. When the metric space is induced by a connected graph, we prove that either there is a line containing all the points or there are \(\Omega (n^{4/7})\) lines, improving the previous \(\Omega (n^{2/7})\) bound. We also prove that the number of lines in an n-point metric space is at least n / 5w, where w is the number of different distances in the space, and we give an \(\Omega (n^{4/3})\) lower bound on the number of lines in metric spaces induced by graphs with constant diameter, as well as spaces where all the positive distances are from {1, 2, 3}.

A constructive version of the Sylvester–Gallai theorem

The Sylvester–Gallai Theorem, stated as a problem by James Joseph Sylvester in 1893, asserts that for any finite, noncollinear set of points on a plane, there exists a line passing through exactly two points of the set. First, it is shown that for the real plane \({{\mathbb{R}^{2}}}\) the theorem is constructively invalid. Then, a well-known classical proof is examined from a constructive standpoint, locating the nonconstructivities. Finally, a constructive version of the theorem is established for the plane \({{\mathbb{R}^{2}}}\); this reveals the hidden constructive content of the classical theorem. The constructive methods used are those proposed by Errett Bishop.

Connectedness of a class of two-dimensional self-affine tiles associated with triangular matrices

We study the connectedness of planar self-affine sets T(A,D) generated by a matrix of the form A=[p0−aq] together with nonconsecutive and noncollinear digit sets of the form D={l0,l1,…,l|p|−1}×{m0,m1,…,m|q|−1}, where {l0,l1,…,l|p|−1} and {m0,m1,…,m|q|−1} are residue systems for |p| and |q| respectively. We give a necessary and sufficient condition for T(A,D) to be connected, and extend some results by Deng and Lau (2011) [5] to nonconsecutive digit sets.

A De Bruijn–Erdős Theorem for Chordal Graphs

A special case of a combinatorial theorem of De Bruijn and Erdős asserts that every noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chávtal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces induced by connected chordal graphs.

Connectedness of planar self-affine sets associated with non-collinear digit sets

We study a class of planar self-affine sets \(T(A,{\mathcal {D}})\) generated by the integer expanding matrices \(A\) with \(|\det A|=3\) and the non-collinear digit sets \({\mathcal {D}}=\{0, v, kAv\}\) where \(k\in {\mathbb {Z}}\setminus \{0\}\) and \(v\in {\mathbb {R}}^{2}\) such that \(\{v, Av\}\) is linearly independent. By examining the characteristic polynomials of \(A\) carefully, we prove that \(T(A,{\mathcal {D}})\) is connected if and only if the parameter \(k=\pm 1\).

The Chen–Chvátal conjecture for metric spaces induced by distance-hereditary graphs

A classical theorem of Euclidean geometry asserts that any noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal conjectured a generalization of this result to arbitrary finite metric spaces, with a particular definition of lines in a metric space. We prove it for metric spaces induced by connected distance-hereditary graphs—a graph G is called distance-hereditary if the distance between two vertices u and v in any connected induced subgraph H of G is equal to the distance between u and v in G.

Progress on Dirac's Conjecture

In 1951, Gabriel Dirac conjectured that every non-collinear set $P$ of $n$ points in the plane contains a point incident to at least $\frac{n}{2}-c$ of the lines determined by $P$, for some constant $c$. The following weakened conjecture was proved by Beck and by Szemerédi and Trotter: every non-collinear set $P$ of $n$ points in the plane contains a point in at least $\frac{n}{c'}$ lines determined by $P$, for some constant $c'$. We prove this result with $c'= 37$. We also give the best known constant for Beck's Theorem, proving that every set of $n$ points with at most $\ell$ collinear determines at least $\frac{1}{98} n(n-\ell)$ lines.

A de Bruijn-Erdős theorem for 1–2 metric spaces

A special case of a combinatorial theorem of De Bruijn and Erdős asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces where each nonzero distance equals 1 or 2.

Torricellian points in normed linear spaces

Abstract Given a set of n (distinct) points A in a normed space, we consider the set of Torricellian points, that is, the set of points which minimises the sum of distances to the points in A . We introduce the Torricellian functional associated to a set of distinct points A , which calculates the sum of distances of a point x to the points in A . The Torricellian point is defined as the infimum (over all vectors) of this functional. We discuss the existence of Torricellian points in reflexive normed spaces, non-expansive subspaces and evidently, inner product spaces. A case for collinear points is given and is utilised to characterise strict convexity. For a non-collinear case, it is shown that the set of Torricellian points contains a unique point when the space is strictly convex. However, we show that the uniqueness of Torricellian point of a non-collinear set does not characterise strict convexity. We consider a particular example of the Torricellian problem in a space endowed with the Taxicab geometry. MSC:46B20, 49J27.

A de Bruijn - Erdos theorem and metric spaces

Combinatorics De Bruijn and Erdos proved that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this theorem in the framework of metric spaces. We provide partial results in this direction.

On an Application of Guth–Katz Theorem

We prove that for some universal $c$, a non-collinear set of $N>\frac{1}{c}$ points in the Euclidean plane determines at least $c \frac{N}{\log N}$ distinct areas of triangles with one vertex at the origin, as well as at least $c \frac{N}{\log N}$ distinct dot products. This in particular implies a sum-product bound $$ |A\cdot A\pm A\cdot A|\geq c\frac{|A|^2}{\log |A|} $$ for a discrete $A \subset {\mathbb R}$.

Disk-like tiles and self-affine curves with noncollinear digits

Let A ∈ M n (ℤ) be an expanding matrix, D C ℤ n a digit set and T = T(A, D) the associated self-affine set. It has been asked by Grochenig and Haas (1994) that given any expanding matrix A ∈ M 2 (Z), whether there exists a digit set such that T is a connected or disk-like (i.e., homeomorphic to the closed unit disk) tile. With regard to this question, collinear digit sets have been studied in the literature. In this paper, we consider noncollinear digit sets and show the existence of a noncollinear digit set corresponding to each expanding matrix such that T is a connected tile. Moreover, for such digit sets, we give necessary and sufficient conditions for T to be a disk-like tile.