# All Questions

0answers
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### $\kappa$-impediments (according to Shelah, Nash-Williams, Aharoni)

Let $\Gamma = (M, W, K)$ be a bipartite graph, that is $M, W$ are sets and $K\subseteq M\times W$. If there is an injective function $f:M\to W$ such that $f\subseteq K$ we say $f$ is an espousal and ...
2answers
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1answer
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1answer
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### Product of posets with Hausdorff interval topology

Given a poset $(P,\leq)$ the interval topology on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = \{y\in P: y\leq x\}$ and ...
1answer
98 views

### When the vertex covering number is smaller than the chromatic number

For any graph $G=(V,E)$ let $\tau(G)$ be the minimum cardinality of a vertex cover of $G$. As noted here, we have $\tau(G) \geq \chi(G) - 1$ for all finite graphs $G$. I'm interested in graphs $G$ ...
2answers
173 views

### “Homomorphism fingerprint” for graphs

Let $G, H$ be simple, undirected graphs without loops. We say that $G, H$ have the same homomorphism fingerprint if $|\text{Hom}(X, G)| = |\text{Hom}(X, H)|$ for all graphs $X$. (By graph ...
2answers
372 views

### Non-isomorphic groups such that there are epis from one to another [closed]

Are there (infinite) non-isomorphic groups $G, H$ such that there are surjective group homomorphisms $f: G\to H$ and $g: H\to G$?
1answer
131 views

### Strongly minimal covers

Let $H=(V,E)$ be a hypergraph, that is $V$ is a set and $E\subseteq \mathcal{P}(V)$. We say that $C\subseteq E$ is a cover of $H$ if $\bigcup C = V$. A cover $M\subseteq E$ is said to be strongly ...
1answer
170 views

### Paracompact zero-dimensional space without clopen partition refinement

If $(X,\tau)$ is a topological space we say that an open cover $\mathcal{U}$ is a clopen partition cover if it consists of disjoint clopen sets. Trivially, every clopen partition cover is locally ...
0answers
67 views

### Linear intersection number and coloring (not chromatic) number

A linear hypergraph is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set and $L\subseteq {\cal P}(X)$ has the following properties: for $e\in L$ we have $|e|\geq 2$; if $e_1\neq e_2 \in L$ then ...
1answer
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### Is it possible to show that $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ diverges?

Let $\mu(n)$ denote the Mobius function with the well-known Dirichlet series representation $$\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}.$$ Basic theorems about Dirichlet series ...
1answer
266 views

### Decidability of Frankl's union-closed sets conjecture

Is it conceivable that Frankl's union closed sets conjecture is undecidable in $\mathsf{ZFC}$, or is this quite implausible, perhaps due to the "finitistic" nature of the statement, or for some other ...
1answer
190 views

### Universal graph homomorphisms

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such ...
1answer
281 views

### Universal maps between topological spaces

Let $X,Y$ be topological spaces. We call a continuous map $u:X\to Y$ universal if for every continous map $f:X\to Y$ there is $x\in X$ such that $f(x) = u(x)$. If $u:X\to Y$ and $v:Y\to Z$ are ...
14answers
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1answer
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### Edge density of triangle-free graphs

Let $G$ be a finite, simple, loopless graph with $|V(G)|=n$. We define its edge density as $$ed(G) := \frac{|E(G)|}{n \choose 2}.$$ Moreover we set d_n := \text{max}\big\{ed(G): G \text{ is a ...
0answers
498 views
+100

### partition of infinite word onto permitted words

Consider words over binary alphabet $\{0,1\}$. Let $M$ be a set of finite words such that $M$ contains at least $c\cdot 2^n$ words of length $n$ for all large enough $n$ (for a constant $c$, ...
1answer
250 views

### Anti-compactness

Let $(X,\tau)$ be a topological space such that $\tau$ is a proper superset of $\{\emptyset, X\}$. We call an open cover $\mathcal{U}$ of $(X,\tau)$ proper if $X\notin \mathcal{U}$. Moreover we say ...
3answers
399 views

### Binary relations as the topological closure of the diagonal

If $(X,\tau)$ is a topological space, we can consider the product topology on $X\times X$ and take the closure of the diagonal $\Delta_X = \{(x,x): x\in X\}$, which we denote by ...
5answers
1k views

### Defining a topology in the Power Set

I have the following question: Given a topological space $T$ is possible in general to give a topology to $2^T$ (the power set of $T$) such that this topology in $2^T$ is related to $T$. If the ...
1answer
877 views

### Did differential geometry undergo a notation change?

As a graduate student, I found the old books of differential geometry used a different set of notation from modern textbooks. For example, Chern and Milnor defined the curvature 2-form by ...
1answer
174 views

### What can be said about graphs if there are homomorphisms in both directions?

Let $G,H$ be simple undirected graphs without loops such that there are graph homomorphisms $f_1:G\to H$ and $f_2: H\to G$. An easy argument shows that we have $\chi(G) = \chi(H)$, even for graphs ...
1answer
201 views

### “Universal maps” as a universal property

In the question Fixed points and universal maps for posets, we find the following definition: if $P, Q$ are partially ordered sets, an order-preserving map $u:P\to Q$ is said to be universal if for ...
1answer
143 views

### How many pairwise non-homeomorphic compact, zero-dimensional topologies are there on $\mathbb{N}$?

To make the question more precise: We call a topological space $(X,\tau)$ zero-dimensional if for $x\neq y \in X$ there is a clopen set $U\subseteq X$ with $x\in U, y\notin U$. Let $\mathcal{C}$ be ...
1answer
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7answers
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### undecidable sentences of first-order arithmetic whose truth values are unknown

Godel's undecidable sentences in first-order arithmetic were guaranteed to be true, by construction. But are there examples of specific sentences known to be undecidable in first-order arithmetic ...
1answer
76 views

### Universal and left-factoring order-preserving maps

Trying to get a different angle for the question Fixed points and universal maps for posets, I want to compare universal maps to a different kind of functions. First recall that for posets $P,Q$ an ...
0answers
147 views

### Does the weak Hadwiger conjecture imply the Hadwiger conjecture?

For any cardinal $\kappa$, let $K_\kappa$ denote the complete graph on $\kappa$. We consider the following statements: (H) If $G$ is a graph and $\chi(G) = \kappa$ then $K_\kappa$ is a minor of $G$. ...
1answer
218 views

### Constructing a function from preimages

This question was inspired by Can we build a continuous function from "fibers"/preimages defined over a topological base? Let $X,Y$ be sets and $L\subseteq \mathcal{P}(Y)$. Suppose $L$ has ...

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