Abstract We prove that for any $$\ell _p$$ ℓ p -norm in the plane with $$1< p< \infty $$ 1 < p < ∞ and for every infinite $$\mathcal {M}\subset \mathbb {R}^2$$ M ⊂ R 2 , there exists a two-colouring of the plane such that no isometric copy of $$\mathcal {M}$$ M is monochromatic. On the contrary, we show that for every polygonal norm (that is, the unit ball is a polygon) in the plane, there exists an infinite $$\mathcal {M}\subset \mathbb {R}^2$$ M ⊂ R 2 such that for every two-colouring of the plane there exists a monochromatic isometric copy of $$\mathcal {M}$$ M .

Abstract. The celebrated Szemerédi–Trotter theorem states that the maximum number of incidences between [Formula: see text] points and [Formula: see text] lines in the plane is [Formula: see text], which is asymptotically tight. Solymosi (2005) conjectured that for any set of points [Formula: see text] and for any set of lines [Formula: see text] in the plane, the maximum number of incidences between [Formula: see text] points and [Formula: see text] lines in the plane whose incidence graph does not contain the incidence graph of [Formula: see text] is [Formula: see text]. This conjecture is mentioned in the book of Brass, et al. (2005) . Even a stronger conjecture, which states that the bound can be improved to [Formula: see text] for some [Formula: see text], was introduced by Mirzaei and Suk (2021) . We disprove both of these conjectures. We also introduce a new approach for proving the upper bound [Formula: see text] on the number of incidences for configurations [Formula: see text] that avoid certain subconfigurations.

Abstract A recent generalization of the Erdős Unit Distance Problem, proposed by Palsson, Senger, and Sheffer, asks for the maximum number of unit distance paths with a given number of vertices in the plane and in 3-space. Studying a variant of this question, we prove sharp bounds on the number of unit distance paths and cycles on the sphere of radius $$1/{\sqrt{2}}$$ 1 / 2 . We also consider a similar problem about 3-regular unit distance graphs in $$\mathbb {R}^3$$ R 3 .

Given a set $S \subseteq \mathbb{R}^2$, define the \emph{Helly number of $S$}, denoted by $H(S)$, as the smallest positive integer $N$, if it exists, for which the following statement is true: for any finite family $\mathcal{F}$ of convex sets in~$\mathbb{R}^2$ such that the intersection of any $N$ or fewer members of~$\mathcal{F}$ contains at least one point of $S$, there is a point of $S$ common to all members of $\mathcal{F}$. We prove that the Helly numbers of \emph{exponential lattices} $\{α^n \colon n \in \mathbb{N}_0\}^2$ are finite for every $α>1$ and we determine their exact values in some instances. In particular, we obtain $H(\{2^n \colon n \in \mathbb{N}_0\}^2)=5$, solving a problem posed by Dillon (2021). For real numbers $α, β> 1$, we also fully characterize exponential lattices $L(α,β) = \{α^n \colon n \in \mathbb{N}_0\} \times \{β^n \colon n \in \mathbb{N}_0\}$ with finite Helly numbers by showing that $H(L(α,β))$ is finite if and only if $\log_α(β)$ is rational.

Abstract In a colouring of $${\mathbb {R}}^d$$ R d a pair $$(S,s_0)$$ ( S , s 0 ) with $$S\subseteq {\mathbb {R}}^d$$ S ⊆ R d and with $$s_0\in S$$ s 0 ∈ S is almost-monochromatic if $$S\setminus \{s_0\}$$ S \ { s 0 } is monochromatic but S is not. We consider questions about finding almost-monochromatic similar copies of pairs $$(S,s_0)$$ ( S , s 0 ) in colourings of $${\mathbb {R}}^d$$ R d , $${\mathbb {Z}}^d$$ Z d , and of $${\mathbb {Q}}$$ Q under some restrictions on the colouring. Among other results, we characterise those $$(S,s_0)$$ ( S , s 0 ) with $$S\subseteq {\mathbb {Z}}$$ S ⊆ Z for which every finite colouring of $${\mathbb {R}}$$ R without an infinite monochromatic arithmetic progression contains an almost-monochromatic similar copy of $$(S,s_0)$$ ( S , s 0 ) . We also show that if $$S\subseteq {\mathbb {Z}}^d$$ S ⊆ Z d and $$s_0$$ s 0 is outside of the convex hull of $$S\setminus \{s_0\}$$ S \ { s 0 } , then every finite colouring of $${\mathbb {R}}^d$$ R d without a monochromatic similar copy of $${\mathbb {Z}}^d$$ Z d contains an almost-monochromatic similar copy of $$(S,s_0)$$ ( S , s 0 ) . Further, we propose an approach based on finding almost-monochromatic sets that might lead to a human-verifiable proof of $$\chi ({{\mathbb {R}}}^2)\ge 5$$

Abstract We say that a set of points $$S\subset {{\mathbb {R}}}^d$$ S ⊂ R d is an $$\varepsilon $$ ε -nearly k-distance set if there exist $$1\le t_1\le \ldots \le t_k$$ 1 ≤ t 1 ≤ … ≤ t k , such that the distance between any two distinct points in S falls into $$[t_1,t_1+\varepsilon ]\cup \cdots \cup [t_k,t_k+\varepsilon ]$$ [ t 1 , t 1 + ε ] ∪ ⋯ ∪ [ t k , t k + ε ] . In this paper, we study the quantity $$\begin{aligned} M_k(d) = \lim _{\varepsilon \rightarrow 0}\max {\{|S|:S\,\text { is an}\, \varepsilon \text {-nearly}\, k\text {-distance set in}\,{{\mathbb {R}}}^d\}} \end{aligned}$$ M k ( d ) = lim ε → 0 max { | S | : S is an ε -nearly k -distance set in R d } and its relation to the classical quantity $$m_k(d)$$ m k ( d ) : the size of the largest k-distance set in $${{\mathbb {R}}}^d$$ R d . We obtain that $$M_k(d)=m_k(d)$$ M k ( d ) = m k ( d ) for $$k=2,3$$ k = 2 , 3 , as well as for any fixed k, provided that d is sufficiently large. The last result answers a question, proposed by Erdős, Makai, and Pach. We also address a closely related Turán-type problem, studied by Erdős, Makai, Pach, and Spencer in the 90s: given n points in $${{\mathbb {R}}}^d$$ R d , how many pairs of them form a distance that belongs to $$[t_1,t_1+1]\cup \cdots \cup [t_k,t_k+1]$$ [

In a colouring of ℝ^d a pair (S,s₀) with S ⊆ ℝ^d and with s₀ ∈ S is almost-monochromatic if S⧵{s₀} is monochromatic but S is not. We consider questions about finding almost-monochromatic similar copies of pairs (S,s₀) in colourings of ℝ^d, ℤ^d, and of ℚ under some restrictions on the colouring. Among other results, we characterise those (S,s₀) with S ⊆ ℤ for which every finite colouring of ℝ without an infinite monochromatic arithmetic progression contains an almost-monochromatic similar copy of (S,s₀). We also show that if S ⊆ ℤ^d and s₀ is outside of the convex hull of S⧵{s₀}, then every finite colouring of ℝ^d without a monochromatic similar copy of ℤ^d contains an almost-monochromatic similar copy of (S,s₀). Further, we propose an approach based on finding almost-monochromatic sets that might lead to a human-verifiable proof of χ(ℝ²) ≥ 5.

Confirming a conjecture of Gyárfás, we prove that, for all natural numbers $k$ and $r$, the vertices of every $r$-edge-colored complete $k$-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for all natural numbers $p$ and $r$, the vertices of every $r$-edge-colored complete graph can be partitioned into a bounded number of $p$th powers of cycles, settling a problem of Elekes, Soukup, Soukup, and Szentmiklóssy [ Discrete Math., 340 (2017), pp. 2053--2069]. In fact we prove a common generalization of both theorems which further extends these results to all host hypergraphs of bounded independence number.