In this paper, we provide a general framework for counting geometric structures in pseudo-random graphs. As applications, our theorems recover and improve several results on the finite field analog of questions originally raised in the continuous setting. The results present interactions between discrete geometry, geometric measure theory, and graph theory.

We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, k-point configuration sets given by

$$ \begin{align*}\Delta^{\mathrm{diag}}(E)= \{ \,|(x,x,\dots,x)-(y_1,y_2,\dots,y_{k-1})| : x, y_1, \dots,y_{k-1} \in E\, \}\end{align*} $$

$E\subset \mathbb {R}^d$

$k\ge 3$

$\Delta ^{\mathrm{diag}}(E)$

(0.1)

$$ \begin{align} \dim(E)> \begin{cases} \frac{2d+1}3, & k=3, \\ \frac{(k-1)d}k,& k\ge 4. \end{cases} \end{align} $$

$C^\omega $

Let $\mathbb {F}_q^d$ denote the d-dimensional vector space over the finite field $\mathbb {F}_q$ with q elements. Define for $\alpha = (\alpha _1, \dots , \alpha _d) \in \mathbb {F}_q^d$. Let $k\in \mathbb {N}$, A be a nonempty subset of $\{(i, j): 1 \leq i < j \leq k + 1\}$ and $r\in (\mathbb {F}_q)^2\setminus {0}$, where $(\mathbb {F}_q)^2=\{a^2:a\in \mathbb {F}_q\}$. If $E\subset \mathbb {F}_q^d$, our main result demonstrates that when the size of the set E satisfies $|E| \geq C_k q^{d/2}$, where $C_k$ is a constant depending solely on k, it is possible to find two $(k+1)$-tuples in E such that one of them is dilated by r with respect to the other, but only along $|A|$ edges. To be more precise, we establish the existence of $(x_1, \dots , x_{k+1}) \in E^{k+1}$ and $(y_1, \dots , y_{k+1}) \in E^{k+1}$ such that, for $(i, j) \in A$, we have $\lVert y_i - y_j \rVert = r \lVert x_i - x_j \rVert $, with the conditions that $x_i \neq x_j$ and $y_i \neq y_j$ for $1 \leq i < j \leq k + 1$, provided that $|E| \geq C_k q^{d/2}$ and $r\in (\mathbb {F}_q)^2\setminus \{0\}$. We provide two distinct proofs of this result. The first uses the technique of group actions, a powerful method for addressing such problems, while the second is based on elementary combinatorial reasoning. Additionally, we establish that in dimension 2, the threshold $d/2$ is sharp when $q \equiv 3 \pmod 4$. As a corollary of the main result, by varying the underlying set A, we determine thresholds for the existence of dilated k-cycles, k-paths and k-stars (where $k \geq 3$) with a dilation ratio of $r\in (\mathbb {F}_q)^2\setminus \{0\}$. These results improve and generalise the findings of Xie and Ge [‘Some results on similar configurations in subsets of $\mathbb {F}_q^d$’, Finite Fields Appl. 91 (2023), Article no. 102252, 20 pages].

Given $E \subseteq \mathbb {F}_q^d \times \mathbb {F}_q^d$ , with the finite field $\mathbb {F}_q$ of order q and the integer $d\,\ge \, 2$ , we define the two-parameter distance set $\Delta _{d, d}(E)=\{(\|x-y\|, \|z-t\|) : (x, z), (y, t) \in E \}$ . Birklbauer and Iosevich [‘A two-parameter finite field Erdős–Falconer distance problem’, Bull. Hellenic Math. Soc. 61 (2017), 21–30] proved that if $|E| \gg q^{{(3d+1)}/{2}}$ , then $ |\Delta _{d, d}(E)| = q^2$ . For $d=2$ , they showed that if $|E| \gg q^{{10}/{3}}$ , then $ |\Delta _{2, 2}(E)| \gg q^2$ . In this paper, we give extensions and improvements of these results. Given the diagonal polynomial $P(x)=\sum _{i=1}^da_ix_i^s\in \mathbb F_q[x_1,\ldots , x_d]$ , the distance induced by P over $\mathbb {F}_q^d$ is $\|x-y\|_s:=P(x-y)$ , with the corresponding distance set $\Delta ^s_{d, d}(E)=\{(\|x-y\|_s, \|z-t\|_s) : (x, z), (y, t) \in E \}$ . We show that if $|E| \gg q^{{(3d+1)}/{2}}$ , then $ |\Delta _{d, d}^s(E)| \gg q^2$ . For $d=2$ and the Euclidean distance, we improve the former result over prime fields by showing that $ |\Delta _{2,2}(E)| \gg p^2$ for $|E| \gg p^{{13}/{4}}$ .

A bipartite graph $H = \left (V_1, V_2; E \right )$ with $\lvert V_1\rvert + \lvert V_2\rvert = n$ is semilinear if $V_i \subseteq \mathbb {R}^{d_i}$ for some $d_i$ and the edge relation E consists of the pairs of points $(x_1, x_2) \in V_1 \times V_2$ satisfying a fixed Boolean combination of s linear equalities and inequalities in $d_1 + d_2$ variables for some s. We show that for a fixed k, the number of edges in a $K_{k,k}$ -free semilinear H is almost linear in n, namely $\lvert E\rvert = O_{s,k,\varepsilon }\left (n^{1+\varepsilon }\right )$ for any $\varepsilon> 0$ ; and more generally, $\lvert E\rvert = O_{s,k,r,\varepsilon }\left (n^{r-1 + \varepsilon }\right )$ for a $K_{k, \dotsc ,k}$ -free semilinear r-partite r-uniform hypergraph.

As an application, we obtain the following incidence bound: given $n_1$ points and $n_2$ open boxes with axis-parallel sides in $\mathbb {R}^d$ such that their incidence graph is $K_{k,k}$ -free, there can be at most $O_{k,\varepsilon }\left (n^{1+\varepsilon }\right )$ incidences. The same bound holds if instead of boxes, one takes polytopes cut out by the translates of an arbitrary fixed finite set of half-spaces.

We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in o-minimal structures (showing that the failure of an almost-linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner).

For two metric spaces $\mathbb X$ and $\mathcal Y$ the chromatic number $\chi ({{\mathbb X}};{{\mathcal{Y}}})$ of $\mathbb X$ with forbidden $\mathcal Y$ is the smallest k such that there is a colouring of the points of $\mathbb X$ with k colors that contains no monochromatic copy of $\mathcal Y$. In this article, we show that for each finite metric space $\mathcal {M}$ that contains at least two points the value $\chi \left ({{\mathbb R}}^n_\infty; \mathcal M \right )$ grows exponentially with n. We also provide explicit lower and upper bounds for some special $\mathcal M$.

Given a finite set $A \subseteq \mathbb{R}^d$, points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$-hole in A if they are the vertices of a convex polytope, which contains no points of A in its interior. We construct arbitrarily large point sets in general position in $\mathbb{R}^d$ having no holes of size $O(4^dd\log d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as (t,m,s)-nets or (t,s)-sequences, yielding a bound of $2^{7d}$. The better bound is obtained using a variant of (t,m,s)-nets, obeying a relaxed equidistribution condition.

Two spheres with centers p and q and signed radii r and s are said to be in contact if |p–q|2=(r–s)2. Using Lie’s line-sphere correspondence, we show that if F is a field in which –1 is not a square, then there is an isomorphism between the set of spheres in F3 and the set of lines in a suitably constructed Heisenberg group that is embedded in (F[i])3; under this isomorphism, contact between spheres translates to incidences between lines.

In the past decade there has been significant progress in understanding the incidence geometry of lines in three space. The contact-incidence isomorphism allows us to translate statements about the incidence geometry of lines into statements about the contact geometry of spheres. This leads to new bounds for Erdős’ repeated distances problem in F3, and improved bounds for the number of point-sphere incidences in three dimensions. These new bounds are sharp for certain ranges of parameters.

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braß–Károlyi–Valtr, Capoyleas–Pach, and Aronov–Dujmovič–Morin–Ooms-da Silveira. We also provide a new variation of the Erdős-Ko-Rado theorem in the ordered setting.

In this paper, we follow and extend a group-theoretic method introduced by Greenleaf–Iosevich–Liu–Palsson (GILP) to study finite points configurations spanned by Borel sets in $\mathbb{R}^n,n\geq 2,n\in\mathbb{N}.$ We remove a technical continuity condition in a GILP’s theorem in [Revista Mat. Iberoamer31 (2015), 799–810]. This allows us to extend the Wolff–Erdogan dimension bound for distance sets to finite points configurations with k points for $k\in\{2,\dots,n+1\}$ forming a $(k-1)$ -simplex.

We show that, for a constant-degree algebraic curve γ in ℝD, every set of n points on γ spans at least Ω(n4/3) distinct distances, unless γ is an algebraic helix, in the sense of Charalambides [2]. This improves the earlier bound Ω(n5/4) of Charalambides [2].

We also show that, for every set P of n points that lie on a d-dimensional constant-degree algebraic variety V in ℝD, there exists a subset S ⊂ P of size at least Ω(n4/(9+12(d−1))), such that S spans $\left({\begin{array}{*{20}{c}} {|S|} \\ 2 \\\end{array}} \right)$ distinct distances. This improves the earlier bound of Ω(n1/(3d)) of Conlon, Fox, Gasarch, Harris, Ulrich and Zbarsky [4].

Both results are consequences of a common technical tool.

We prove that n plane algebraic curves determine O(n(k+2)/(k+1)) points of kth order tangency. This generalizes an earlier result of Ellenberg, Solymosi and Zahl on the number of (first order) tangencies determined by n plane algebraic curves.

For each $n\geqslant 2$ we construct a measurable subset of the unit ball in $\mathbb{R}^{n}$ that does not contain pairs of points at distance 1 and whose volume is greater than $(1/2)^{n}$ times the volume of the unit ball. This disproves a conjecture of Larman and Rogers from 1972.

In this paper we discuss some dimension results for triangle sets of compact sets in $\mathbb{R}^{2}$. In particular we prove that for any compact set $F$ in $\mathbb{R}^{2}$, the triangle set $\unicode[STIX]{x1D6E5}(F)$ satisfies

$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant {\textstyle \frac{3}{2}}\dim _{\text{A}}F.\end{eqnarray}$$

$\dim _{\text{A}}F>1$

$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant 1+\dim _{\text{A}}F.\end{eqnarray}$$

$\dim _{\text{A}}F>4/3$

$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant \min \{{\textstyle \frac{5}{2}}\dim _{\text{A}}F-1,3\}.\end{eqnarray}$$

$F$

$$\begin{eqnarray}\text{}\underline{\dim _{\text{B}}}F\geqslant {\textstyle \frac{3}{2}}\text{}\underline{\dim _{\text{B}}}\unicode[STIX]{x1D6E5}(F)\quad \text{and}\quad \overline{\dim _{\text{B}}}F\geqslant {\textstyle \frac{3}{2}}\overline{\dim _{\text{B}}}\unicode[STIX]{x1D6E5}(F).\end{eqnarray}$$

A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least (1−o(1))n 2.

We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.)

An important ingredient of our proofs is the following statement. Let S be a family of n open curves in ℝ2, so that each curve is the graph of a continuous real function defined on ℝ, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is $\Omega(nt\sqrt{\log t/\log\log t})$ .

We show an equivalence between a conjecture of Bisztriczky and Fejes Tóth about families of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this equivalence we improve the upper bound of Pach and Tóth on the Erdős–Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk on the Erdős–Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erdős–Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erdős–Szekeres theorem of Pór and Valtr to families of non-crossing convex bodies.

An intersection graph of curves in the plane is called a string graph. Matoušek almost completely settled a conjecture of the authors by showing that every string graph with m edges admits a vertex separator of size $O(\sqrt{m}\log m)$. In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphs G with n vertices: (i) if Kt ⊈ G for some t, then the chromatic number of G is at most (log n)O(log t); (ii) if Kt,t ⊈ G, then G has at most t(log t)O(1)n edges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdős–Hajnal conjecture almost holds for string graphs.

For three points , and in the n-dimensional space 𝔽nq over the finite field 𝔽q of q elements we give a natural interpretation of an acute angle triangle defined by these points. We obtain an upper bound on the size of a set 𝒵 such that all triples of distinct points define acute angle triangles. A similar question in the real space ℛn dates back to P. Erdős and has been studied by several authors.