Nonempty Intersection Research Articles

Coloring Hypergraphs Defined by Stabbed Pseudo-Disks and ABAB-Free Hypergraphs

What is the minimum number of colors that always suffice to color every planar set of points such that any disk that contains enough points contains two points of different colors? It is known that the answer to this question is either three or four. We show that three colors always suffice if the condition must be satisfied only by disks that contain a fixed point. Our result also holds, and is even tight, when instead of disks we consider their topological generalization, namely, pseudo-disks, with a nonempty intersection. Our solution uses the equivalence that a hypergraph can be realized by stabbed pseudo-disks if and only if it is ABAB-free. These hypergraphs are defined in a purely abstract, combinatorial way, and our proof that they are 3-chromatic is also combinatorial.

Examples of Sierpiński–Zygmund maps in the class of Darboux-like functions

The Darboux-like functions represent a group of maps that are continuous in a generalized sense. The algebra of subsets of $${\mathbb {R}}^{\mathbb {R}}$$ (i.e., maps from $${\mathbb {R}}$$ to $${\mathbb {R}}$$) generated by these classes has nine atoms, that is, the smallest non-empty elements of the algebra. The subject of this work is to study the intersections of these atoms with the class $${{\,\mathrm{SZ}\,}}$$ of Sierpinski–Zygmund functions—the maps that have as little of the standard continuity as possible. Specifically, we will show that it is independent of the standard axioms of set theory that each of these atoms has a non-empty intersection with $${{\,\mathrm{SZ}\,}}$$. For seven of the nine atoms this has been unknown, and the constructions of the examples provide answers to the problems stated in a recent survey A century of Sierpinski–Zygmund functions of K. C. Ciesielski and J. Seoane-Sepulveda. Notice that lineability of the main classes of Darboux-like functions, as well as of Sierpinski–Zygmund functions, has been intensively studied. The presented work opens a possibility to study also the lineability of the nine smaller classes we discuss here.

Comparing Degenerate Strings

Uncertain sequences are compact representations of sets of similar strings. They highlight common segments by collapsing them, and explicitly represent varying segments by listing all possible options. A generalized degenerate string (GD string) is a type of uncertain sequence. Formally, a GD string Ŝ is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length ki but this length can vary between different sets. We denote by W the sum of these lengths k0, k1, . . . , kn-1. Our main result is an 𝒪(N + M)-time algorithm for deciding whether two GD strings of total sizes N and M, respectively, over an integer alphabet, have a non-empty intersection. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in linear space. We then apply our string comparison tool to devise a simple algorithm for computing all palindromes in Ŝ in 𝒪(min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in Ŝ. We also show that a result, which is essentially the same as our string comparison linear-time algorithm, can be obtained by employing an automata-based approach.

A Numerical Approach for the Filtered Generalized Čech Complex

In this paper, we present an algorithm to compute the filtered generalized Čech complex for a finite collection of disks in the plane, which do not necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris–Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments. We give an algorithm for computing the 2-dimensional filtered generalized Čech complex of a finite collection of d-dimensional disks in R d , and we show the performance of our algorithm.

Stability of a uniform attractor for a second-order evolution equation with discontinuous trajectories

The work is devoted to the study of the qualitative behavior of solutions of the wave equation, whose trajectories have impulsive perturbations in moments when they reach a fixed (impulsive) subset of the phase space. Using the general constructing scheme of the infinite-dimensional impulsive dynamical system and using the concept of a uniform attractor — a minimal compact uniformly attracting set, we obtained the result about the existence and the explicit form of a uniform attractor for the corresponding impulsive dynamical system. Trajectories of such system can have an infinite number of impulsive points when they encounter an impulsive subset of the phase space. Thus, a uniform attractor may have a non-empty intersection with the impulsive set and, as a result, may not have the stability property. However, due to the additional conditions on the impulsive parameters of the problem, we managed to prove the stability property for the non-impulsive part of the uniform attractor.

HYPERHROUPS AND Hv-GROUPS ASSOCIATED TO ELEMENTS WITH FOUR OXIDATION STATES

The theory of hyperstructures is of great importance due to its connections to various fields of Science. $H_v$-structures are hyperstructures where the equality is replaced by the nonempty intersection. This class of the hyperstructures is very large so one can use it in order to define several objects that they are not possible to be defined in the classical hyperstructure theory. This paper attempts an exposition of the connection between hyperstructure ($H_v$-structure) theory and certain type of chemical reactions. In this regard, we consider elements with four oxidation states and investigate their mathematical structures.

Incompatible Intersection Properties

Let F ⊂ 2 [n] be a family in which any three sets have non-empty intersection and any two sets have at least 32 elements in common. The nearly best possible bound F ≤ 2n−2 is proved. We believe that 32 can be replaced by 3 and provide a simple-looking conjecture that would imply this.

Bounds on upper transversals in hypergraphs

A set S of vertices in a hypergraph H is a transversal if it has a nonempty intersection with every edge of H. For $$k \ge 1$$, if H is a hypergraph with every edge of size at least k, then a k-transversal in H is a transversal that intersects every edge of H in at least k vertices. In particular, a 1-transversal is a transversal. The upper k-transversal number $$\Upsilon _{k}(H)$$ of H is the maximum cardinality of a minimal k-transversal in H. Let H be a hypergraph with $$n_{_H}$$ vertices and $$m_{_H}$$ edges. We show that for $$r \ge 2$$ and for every integer $$k \in [r]$$, if H is r-uniform with maximum degree $$\Delta $$, then $$\Upsilon _{k}(H) \le \left( \frac{k \cdot \Delta }{k (\Delta - 1) + r} \right) n_{_H}$$ and $$\Upsilon _{k}(H) \le \left( \frac{k \cdot \Delta }{\Delta (k + 1) + r - k} \right) (n_{_H}+ m_{_H})$$, and both bounds are tight. As a special case of this result, if H is a 3-regular, 3-uniform hypergraph, then $$\Upsilon _{2}(H) \le \frac{6}{7} n_{_H}$$, and equality in this bound is achieved by the Fano plane. We also discuss a relation between upper transversals in 3-uniform hypergraphs and the famous cap set problem, and show that for every given $$\epsilon > 0$$, there exists a 3-uniform, connected, linear hypergraphs of sufficiently large order such that $$\Upsilon _{1}(H) < \epsilon \cdot n_{_H}$$.

Comparing Two Clusterings Using Matchings between Clusters of Clusters

Clustering is a fundamental problem in data science, yet the variety of clustering methods and their sensitivity to parameters make clustering hard. To analyze the stability of a given clustering algorithm while varying its parameters, and to compare clusters yielded by different algorithms, several comparison schemes based on matchings, information theory, and various indices (Rand, Jaccard) have been developed. We go beyond these by providing a novel class of methods computing meta-clusters within each clustering—a meta-cluster is a group of clusters, together with a matching between these. Let the intersection graph of two clusterings be the edge-weighted bipartite graph in which the nodes represent the clusters, the edges represent the nonempty intersection between two clusters, and the weight of an edge is the number of common items. We introduce the so-called D -family-matching problem on intersection graphs, with D the upper bound on the diameter of the graph induced by the clusters of any meta-cluster. First we prove NP -completeness and APX -hardness results, and unbounded approximation ratio of simple strategies. Second, we design exact polynomial time dynamic programming algorithms for some classes of graphs (in particular trees). Then we prove spanning tree–based efficient heuristic algorithms for general graphs. Our experiments illustrate the role of D as a scale parameter providing information on the relationship between clusters within a clustering and in-between two clusterings. They also show the advantages of our built-in mapping over classical cluster comparison measures such as the variation of information.

Mixed matchings in graphs

Let n,s be integers, n≥2(s+1)≥4. Let F,G⊂[n]2 be two graphs. We determine the exact maximum of |F|+|G| subject to the condition that there is no matching of size s+1 contained in F∪G and having non-empty intersection with both F and G.

Probabilistic pursuits on graphs

We consider discrete dynamical systems of “ant-like” agents engaged in a sequence of pursuits on a graph environment. The agents emerge one by one at equal time intervals from a source vertex s and pursue each other by greedily attempting to close the distance to their immediate predecessor, the agent that emerged just before them from s, until they arrive at the destination point t. Such pursuits have been investigated before in the continuous setting and in discrete time when the underlying environment is a regular grid. In both these settings the agents' walks provably converge to a shortest path from s to t. Furthermore, assuming a certain natural probability distribution over the move choices of the agents on the grid (in case there are multiple shortest paths between an agent and its predecessor), the walks converge to the uniform distribution over all shortest paths from s to t - and so the agents' locations are on average very close to the straight line from s to t.In this work we study the evolution of agent walks over a general finite graph environment G. Our model is a natural generalization of the pursuit rule proposed for the case of the grid. The main results are as follows. We show that “convergence” to the shortest paths in the sense of previous work extends to all pseudo-modular graphs (i.e. graphs in which every three pairwise intersecting disks have a nonempty intersection), and also to environments obtained by taking graph products, generalizing previous results in two different ways. We show that convergence to the shortest paths is also obtained by chordal graphs (i.e. graphs in which all cycles of four or more vertices have a chord), and discuss some further positive and negative results for planar graphs. In the most general case, convergence to the shortest paths is not guaranteed, and the agents may get stuck on sets of recurrent, non-optimal walks from s to t. However, we show that the limiting distributions of the agents' walks will always be uniform distributions over some set of walks of equal length.

A complex Gap lemma

Inspired by the work of Newhouse in one real variable, we introduce a relevant notion of thickness for dynamical Cantor sets in the plane associated to a holomorphic IFS. Our main result is a complex version of Newhouse’s Gap Lemma: we show that under some assumptions, if the product of the thicknesses of two Cantor sets K K and L L is larger than 1, then K K and L L have non-empty intersection. Since in addition this thickness varies continuously, this gives a criterion to get a robust intersection between two Cantor sets in the plane.

Two infinite families of critical clique–Helly graphs

A graph is clique–Helly if every family of pairwise intersecting (maximal) cliques has non-empty total intersection. Dourado, Protti and Szwarcfiter conjectured that every clique–Helly graph contains a vertex whose removal maintains it as a clique–Helly graph. We present here two infinite families of counterexamples to this conjecture.

Intersection graphs associated with semigroup acts

The intersection graph $\mathbb{Int}(A)$ of an $S$-act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are non-trivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a non-empty intersection. In this paper, we study some graph-theoretic properties of $\mathbb{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the finiteness of each of the clique number, the chromatic number, and the degree of some or all vertices in $\mathbb{Int}(A)$ is equivalent to the finiteness of the number of subacts of $A$. Finally, we determine the clique number of the graphs of certain classes of $S$-acts.

On the union of intersecting families

AbstractA family of sets is said to be intersecting if any two sets in the family have non-empty intersection. In 1973, Erdős raised the problem of determining the maximum possible size of a union of r different intersecting families of k-element subsets of an n-element set, for each triple of integers (n, k, r). We make progress on this problem, proving that for any fixed integer r ⩾ 2 and for any $$k \le ({1 \over 2} - o(1))n$$, if X is an n-element set, and $${\cal F} = {\cal F}_1 \cup {\cal F}_2 \cup \cdots \cup {\cal F}_r $$, where each $$ {\cal F}_i $$ is an intersecting family of k-element subsets of X, then $$|{\cal F}| \le \left( {\matrix{n \cr k \cr } } \right) - \left( {\matrix{{n - r} \cr k \cr } } \right)$$, with equality only if $${\cal F} = \{ S \subset X:|S| = k,\;S \cap R \ne \emptyset \} $$ for some R ⊂ X with |R| = r. This is best possible up to the size of the o(1) term, and improves a 1987 result of Frankl and Füredi, who obtained the same conclusion under the stronger hypothesis $$k &amp;#x003C; (3 - \sqrt 5 )n/2$$, in the case r = 2. Our proof utilizes an isoperimetric, influence-based method recently developed by Keller and the authors.

Completely distributive completions of posets

A $$\Delta_1$$ -completion of a poset is a completion for which, simultaneously, every element is reachable as a join of meets and a meet of joins from the original poset. We focus our attention on $$\Delta_1$$ -completions that can be obtained from polarities $$\langle\mathcal{F},\mathcal{I},\mathcal{R}\rangle$$ where $$\mathcal{F}$$ is a collection of upsets containing the principal upsets, $$\mathcal{I}$$ is a collection of downsets containing the principal downsets of the original poset, and $$R\subseteq\mathcal{F}\times\mathcal{I}$$ is the relation of nonempty intersection. These $$\Delta_1$$ -completions are called $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions, and they satisfy a compactness property. In this paper, we show that if a pair $$\langle\mathcal{F},\mathcal{I}\rangle$$ satisfies a separating condition (similar to the Prime Filter Theorem for distributive lattices), then the $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completion of the original poset is a completely distributive algebraic lattice. Given a poset P and an algebraic closure system $$\mathcal{F}$$ of upsets of P satisfying a distributivity condition, we show how to choose a collection of downsets $$\mathcal{I}$$ of P such that the $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completion of P is a completely distributive algebraic lattice. Then, we study the extensions of additional operations on posets to their corresponding $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions. Finally, we use the previous results to obtain adequate $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions for the classes of Tarski algebras and Hilbert algebras, and for the classes of algebras that are canonically associated (in the sense of abstract algebraic logic) with some propositional logics.

On the positive geometry of conformal field theory

It has long been clear that the conformal bootstrap is associated with a rich geometry. In this paper we undertake a systematic exploration of this geometric structure as an object of study in its own right. We study conformal blocks for the minimal SL(2, R) symmetry present in conformal field theories in all dimensions. Unitarity demands that the Taylor coefficients of the four-point function lie inside a polytope U determined by the operator spectrum, while crossing demands they lie on a plane X. The conformal bootstrap is then geometrically interpreted as demanding a non-empty intersection of U ∩ X. We find that the conformal blocks enjoy a surprising positive determinant property. This implies that U is an example of a famous polytope — the cyclic polytope. The face structure of cyclic polytopes is completely understood. This lets us fully characterize the intersection U∩X by a simple combinatorial rule, leading to a number of new exact statements about the spectrum and four-point function in any conformal field theory.

Further Consequences of the Colorful Helly Hypothesis

Let \(\mathcal {F}\) be a family of convex sets in \({\mathbb {R}}^d,\) which are colored with \(d+1\) colors. We say that \(\mathcal {F}\) satisfies the Colorful Helly Property if every rainbow selection of \(d+1\) 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 \(\mathcal {F}\) there is a color class \(\mathcal {F}_i\subset \mathcal {F},\) for \(1\le i\le d+1,\) whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension \(d\ge 2\) 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 \(\mathcal {F}\) can be crossed by g(d) lines.

Further Results on Existentially Closed Graphs Arising from Block Designs

A graph is $n$-existentially closed ($n$-e.c.) if for any disjoint subsets $A$, $B$ of vertices with $|{A \cup B}|=n$, there is a vertex $z \notin A \cup B$ adjacent to every vertex of $A$ and no vertex of $B$. For a block design with block set $\cal B$, its block intersection graph is the graph whose vertex set is $\cal B$ and two vertices (blocks) are adjacent if they have non-empty intersection. In this paper, we investigate the block intersection graphs of pairwise balanced designs, and propose a sufficient condition for such graphs to be $2$-e.c. In particular, we study the $\lambda$-fold triple systems with $\lambda \ge 2$ and determine for which parameters their block intersection graphs are $1$- or $2$-e.c. Moreover, for Steiner quadruple systems, the block intersection graphs and their analogue called $\{1\}$-block intersection graphs are investigated, and the necessary and sufficient conditions for such graphs to be $2$-e.c. are established.

Polygon-circle and word-representable graphs

We describe work on the relationship between the independently-studied polygon-circle graphs and word-representable graphs.A graph G = (V, E) is word-representable if there exists a word w over the alpha-bet V such that letters x and y form a subword of the form xyxy ⋯ or yxyx ⋯ iff xy is an edge in E. Word-representable graphs generalise several well-known and well-studied classes of graphs [S. Kitaev, A Comprehensive Introduction to the Theory of Word-Representable Graphs, Lecture Notes in Computer Science 10396 (2017) 36–67; S. Kitaev, V. Lozin, “Words and Graphs”, Springer, 2015]. It is known that any word-representable graph is k-word-representable, that is, can be represented by a word having exactly k copies of each letter for some k dependent on the graph. Recognising whether a graph is word-representable is NP-complete ([S. Kitaev, V. Lozin, “Words and Graphs”, Springer, 2015, Theorem 4.2.15]). A polygon-circle graph (also known as a spider graph) is the intersection graph of a set of polygons inscribed in a circle [M. Koebe, On a new class of intersection graphs, Ann. Discrete Math. (1992) 141–143]. That is, two vertices of a graph are adjacent if their respective polygons have a non-empty intersection, and the set of polygons that correspond to vertices in this way are said to represent the graph. Recognising whether an input graph is a polygon-circle graph is NP-complete [M. Pergel, Recognition of polygon-circle graphs and graphs of interval filaments is NP-complete, Graph-Theoretic Concepts in Computer Science: 33rd Int. Workshop, Lecture Notes in Computer Science, 4769 (2007) 238–247]. We show that neither of these two classes is included in the other one by showing that the word-representable Petersen graph and crown graphs are not polygon-circle, while the non-word-representable wheel graph W5 is polygon-circle. We also provide a more refined result showing that for any k ≥ 3, there are k-word-representable graphs which are neither (k −1)-word-representable nor polygon-circle.