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Dominic van der Zypen

Publications and source records attributed to Dominic van der Zypen.

At least 19 recordsLinked to original sources
arXiv2026

Hadwiger's conjecture for hypergraphs

In 1943, Hadwiger formulated his celebrated conjecture, connecting the chromatic number $χ(G)$ of a finite, simple, undirected graph with the cardinality of the largest complete minor, $η(G)$. The disjoint union of all finite complete graphs shows that Hadwiger's conjecture fails for infinite, but a slightly weaker version is true in these graphs, and open for finite graphs. In this note we generalize that weaker version to hypergraphs and provide a simple, general, and purely set-theoretical formulation of Hadwiger's conjecture.

math.LO↗
arXiv2026

The matroid intersection conjecture

We prove the matroid intersection conjecture, due to Nash-Williams \cite{ahaziv}, for infinite matroids: If $M_k = (E, {\mathcal I}_k)$ are matroids for $k \in \{0,1\}$, both having the same base set $E$, then there is $J^* \in {\mathcal I}_0 \cap {\mathcal I}_1$ and disjoint subsets $J_k\subseteq J^*$ for $k\in \{0,1\}$ such that $\text{cl}_{M_0}(J_0) \cup \text{cl}_{M_1}(J_1) = E$.

math.CO↗
arXiv2025

${Ω\choose T}\neq {Ω\choose Γ}$

We show that the selection principles ${Ω\choose T}$ and ${Ω\chooseΓ}$ are not equal constructing a topological space $(X,τ)$ that satisfies ${Ω\choose T}$, but not ${Ω\choose Γ}$. This answers a question from arXiv:math/0301011 .

math.GN↗
arXiv2024

Heritability of Kőnig's Property from finite edge sets

A hypergraph $H = (V,E)$ is said to have Kőnig's Property if there is a matching $M\subseteq E$ and $S\subseteq V$ such that $|S \cap e| = 1$ for all $e\in M$, and $S$ is a vertex cover of $H$. Aharoni posed the question whether Kőnig's Property is inheritable from finite subsets of $E$. We provide a negative answer and investigate similar questions for weaker properties.

math.CO↗
arXiv2023

Knuth's non-associative "group" on ${\mathcal P}(\mathbb{N})$

Donald Knuth introduced in The Art of Computer Programming (Vol 4a) a fast approximation to the addition of integers (given in binary) in terms of bit-wise operations by $a + b \; \approx \; a \oplus b \oplus ((a\land b) \ll 1).$ Generalizing this to infinite bit-strings we get a binary operation on ${\mathcal P}(\mathbb{N})$, the power-set of $\mathbb{N}$ (which we identify with the collection of infinite bit-strings). We show that this operation is ``group-like'' in that it has a neutral element, inverses, but it is not associative. There are a lot of questions left, which the author has not been able to answer.

math.GM↗
arXiv2022

Graph embeddings into Hamming spaces

Graph embeddings deal with injective maps from a given simple, undirected graph $G=(V,E)$ into a metric space, such as $\mathbb{R}^n$ with the Euclidean metric. This concept is widely studied in computer science, see \cite{ge1}, but also offers attractive research in pure graph theory \cite{ge2}. In this note we show that any graph can be embedded into a particularly simple metric space: $\{0,1\}^n$ with the Hamming distance, for large enough $n$.

math.CO↗
arXiv2021

On a Question of Grätzer and Lakser from the 1971 {\sl Transactions of the American Mathematical Society}

Grätzer and Lakser asked in the 1971 {\sl Transactions of the American Mathematical Society} if the pseudocomplemented distributive lattices in the amalgamation class of the subvariety generated by ${\bf 2}^n\oplus{\bf 1}$ can be characterized by the property of not having a $*$-homomorphism onto ${\bf 2}^i\oplus{\bf 1}$ for $1<i<n$. In this article, this question is answered. If you want to know the answer, you will have to read it (or skip to the last section).

math.CO↗
arXiv2018

Minimal covers of hypergraphs

For a hypergraph $H=(V,\mathcal E)$, a subfamily $\mathcal C\subseteq \mathcal E$ is called a cover of the hypergraph if $\bigcup\mathcal C=\bigcup\mathcal E$. A cover $\mathcal C$ is called minimal if each cover $\mathcal D\subseteq\mathcal C$ of the hypergraph $H$ coincides with $\mathcal C$. We prove that for a hypergraph $H$ the following conditions are equivalent: (i) each countable subhypergraph of $H$ has a minimal cover; (ii) each non-empty subhypergraph of $H$ has a maximal edge; (iii) $H$ contains no isomorphic copy of the hypergraph $(ω,ω)$. This characterization implies that a countable hypergraph $(V,\mathcal E)$ has a minimal cover if every infinite set $I\subseteq V$ contains a finite subset $F\subseteq I$ such that the family of edges $\mathcal E_F:=\{E\in\mathcal E:F\subseteq E\}$ is finite. Also we prove that a hypergraph $(V,\mathcal E)$ has a minimal cover if $\sup\{|E|:E\in\mathcal E\}<ω$ or for every $v\in V$ the family $\mathcal E_v:=\{E\in\mathcal E:v\in E\}$ is finite.

math.CO↗
arXiv2016

Order and interval topologies on complete Boolean algebras

We introduce and examine order convergence and the interval topology on partially ordered sets in general. Problem 76 of Birkhoff's "Lattice Theory" asks whether for complete Boolean algebras the order topology and the interval topology coincide. We answer this question in the negative.

math.LO↗
arXiv2016

Majority Colourings of Digraphs

We prove that every digraph has a vertex 4-colouring such that for each vertex $v$, at most half the out-neighbours of $v$ receive the same colour as $v$. We then obtain several results related to the conjecture obtained by replacing 4 by 3.

math.CO↗
arXiv2015

Graph-theoretic autofill

Imagine a website that asks the user to fill in a web form and -- based on the input values -- derives a relevant figure, for instance an expected salary, a medical diagnosis or the market value of a house. How to deal with missing input values at run-time? Besides using fixed defaults, a more sophisticated approach is to use predefined dependencies (logical or correlational) between different fields to autofill missing values in an iterative way. Directed loopless graphs (in which cycles are allowed) are the ideal mathematical model to formalize these dependencies. We present two new graph-theoretic approaches to filling missing values at run-time.

cs.HC↗
arXiv2015

On incomplete lattice homomorphisms in subspaces of geometries: "half" a problem of Hartmanis from 1959

Turing Award winner Juris Hartmanis introduced in 1959 lattices of subspaces of generalized partitions ("partitions of type n"; "geometries" if $n = 2$). Hartmanis states it is "an unsolved problem whether there are any incomplete lattice homomorphisms in" lattices of subspaces of geometries. (He continues, "[I]f so how can these geometries be characterized.") We give a positive answer to this question.

math.LO↗
arXiv2013

A weak form of Hadwiger's conjecture

We introduce the following weak version of Hadwiger's conjecture: If $G$ is a graph and $κ$ is a cardinal such that there is no coloring map $c:G \to κ$, then $K_κ$ is a minor of $G$. We prove that this statement is true for graphs with infinite chromatic number

math.CO↗