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Person# Bartosz Maria Walczak

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Related research domains (5)

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one

Rights

Rights are legal, social, or ethical principles of freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal

Intersection graph

In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special c

Related publications (9)

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János Pach, Bartosz Maria Walczak

Suppose k is a positive integer and X is a k-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most k sets. Suppose there is a function f(n) = o(n(2)) with the property that any n members of X determine at most f(n) holes, which means that the complement of their union has at most f(n) bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that X can be decomposed into at most p (1-fold) packings, where p is a constant depending only on k and f.

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For positive integers w and k, two vectors A and B from Z(w) are called k-crossing if there are two coordinates i and j such that A[i] - B[i] >= k and B[j] - A[j] >= k. What is the maximum size of a family of pairwise 1-crossing and pairwise non-k-crossing vectors in Z(w)? We state a conjecture that the answer is k(w-1). We prove the conjecture for w = 4. Also, for all k and so, we construct several quite different examples of families of desired size k(w-1). This research is motivated by a natural question concerning the width of the lattice of maximum antichains of a partially ordered set. (C) 2019 Elsevier Inc. All rights reserved.

Michal Lason, Bartosz Maria Walczak

A family of sets in the plane is simple if the intersection of any subfamily is arc-connected, and it is pierced by a line L if the intersection of any member with L is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their clique number.