In this work we study geometric realizations of permutations of point sets in the plane under strict non-crossing constraints. Specifically, we consider non-self-intersecting paths over sets of n + 1 points in general position with distinct y -coordinates. Each such path has a signature (a word in {-, +} n ), describing the vertical movement along the path. A signature is always-realizable if it can be realized by a non-self-intersecting path for every such point set. A well-known conjecture in upward planar embedding problems is that every signature is always-realizable. We provide new constructive methods that yield broad families of such signatures, including all signatures of length up to 9. We introduce the concept of forward-convex paths, which serves as a key tool for algorithmic path construction. Our framework allows for systematic composition of signature fragments, enabling general constructions. Our results generalize and unify previous work on upward path embeddings onto arbitrary point sets.

Abstract Let K be a compact convex set in $$\mathbb {R}^{2}$$ R 2 and let $$\mathcal {F}_{1}, \mathcal {F}_{2}, \mathcal {F}_{3}$$ F 1 , F 2 , F 3 be finite families of translates of K such that $$A \cap B \ne \emptyset $$ A ∩ B ≠ ∅ for every $$A \in \mathcal {F}_{i}$$ A ∈ F i and $$B \in \mathcal {F}_{j}$$ B ∈ F j with $$i \ne j$$ i ≠ j . A conjecture by Dol’nikov is that, under these conditions, there is always some $$j \in \{ 1,2,3 \}$$ j ∈ { 1 , 2 , 3 } such that $$\mathcal {F}_{j}$$ F j can be pierced by 3 points. In this paper we prove a stronger version of this conjecture when K is a body of constant width or when it is close in Banach-Mazur distance to a disk. We also show that the conjecture is true with 8 piercing points instead of 3. Along the way we prove more general statements both in the plane and in higher dimensions. A related result was given by Martínez-Sandoval, Roldán-Pensado and Rubin. They showed that if $$\mathcal {F}_{1}, \dots , \mathcal {F}_{d}$$ F 1 , ⋯ , F d are finite families of convex sets in $$\mathbb {R}^{d}$$ R d such that for every choice of sets $$C_{1} \in \mathcal {F}_{1}, \dots , C_{d} \in \mathcal {F}_{d}$$ C 1 ∈ F 1 , ⋯ , C d ∈ F d the intersection $$\bigcap _{i=1}^{d} {C_{i}}$$ ⋂ i = 1 d C i is non-empty, then either there exists $$j \in \{ 1,2, \dots , n \}$$ j ∈ { 1 , 2 , ⋯ , n } such that $$\mathcal {F}_j$$

In this note we give a negative answer to a question proposed by Almendra-Hernández and Martínez-Sandoval. Let n ≤ m be positive integers and let X and Y be sets of sizes n and m in Rn-1 such that X U Y is in generic position. There is a natural order on X x Y induced by the distances between the corresponding points. The question is if all possible orders on X x Y can be obtained in this way. We show that the answer is negative when n < m. The case n=m remains open.

Let $$\mathcal {F}$$ F be a family of convex sets in $${\mathbb {R}}^d,$$ R d , which are colored with $$d+1$$ d + 1 colors. We say that $$\mathcal {F}$$ F satisfies the Colorful Helly Property if every rainbow selection of $$d+1$$ d + 1 sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovász states that for any such colorful family $$\mathcal {F}$$ F there is a color class $$\mathcal {F}_i\subset \mathcal {F},$$ F i ⊂ F , for $$1\le i\le d+1,$$ 1 ≤ i ≤ d + 1 , whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension $$d\ge 2$$ d ≥ 2 there exist numbers f(d) and g(d) with the following property: either one can find an additional color class whose sets can be pierced by f(d) points, or all the sets in $$\mathcal {F}$$ F can be crossed by g(d) lines.

We continue a sequence of recent works studying Ramsey functions for semialgebraic predicates in Rd. A k-ary semialgebraic predicate Φ(x1, …, xk) on Rd is a Boolean combination of polynomial equations and inequalities in the kd coordinates of k points x1, …, xk ∈ Rd. A sequence P = (p1, …, pn) of points in Rd is called Φ-homogeneous if either Φ(pi1, …,pik) holds for all choices 1 ≤ i1 < … < ik ≤ n, or it holds for no such choice. The Ramsey function RΦ (n) is the smallest N such that every point sequence of length N contains a Φ-homogeneous subsequence of length n.

We continue a sequence of recent works studying Ramsey functions for semialgebraic predicates in $\mathbb{R}^d$. A $k$-ary semialgebraic predicate $\Phi(x_1,\ldots,x_k)$ on $\mathbb{R}^d$ is a Boolean combination of polynomial equations and inequalities in the $kd$ coordinates of $k$ points $x_1,\ldots,x_k\in\mathbb{R}^d$. A sequence $P=(p_1,\ldots,p_n)$ of points in $\mathbb{R}^d$ is called $\Phi$-homogeneous if either $\Phi(p_{i_1}, \ldots,p_{i_k})$ holds for all choices $1\le i_1<\cdots

In his paper "On sets of distances of n points", Paul Erdos conjectured that every convex curve contains a point P such that every circle centered at P intersects the curve in at most 2 points. This conjecture is false: If T is an equilateral triangle with boundary T, for any point P on T there is a circle centered at P that intersects T at 4 points. But perhaps the number 2 in Erdos's conjecture can be replaced by some other number.