22 votes
Accepted

Minimal good cover of the torus

You can't do any better than $7$. This follows from Karoubi, Max; Weibel, Charles A., On the covering type of a space, Enseign. Math. (2) 62, No. 3-4, 457-474 (2016). ZBL1378.55002. in particular ...
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  • 33.3k
18 votes
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Are there infinitely many natural numbers not covered by one of these 7 polynomials?

Notice that $$30\cdot f_1(n_1,m_1) = (30\cdot m_1+23)\cdot (30\cdot n_1+7) - 11\\ 30\cdot f_2(n_2,m_2) = (30\cdot m_2+17)\cdot (30\cdot n_2+13) - 11\\ 30\cdot f_3(n_3,m_3) = (30\cdot m_3+23)\cdot (30\...
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13 votes

Normal Covering of a Finite Group

The following paper is relevant: Finite coverings by normal subroups by Brodie, Chamberlain and Kapp, PAMS 1988. The main focus is on coverings of infinite groups although their main theorem is ...
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  • 10.8k
11 votes
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Image of curve along a finite etale Galois map

No. Just take two points in $X$ that map to the same point $p$ in $Y$; then a general curve in $X$ containing these two points will have image with a node at $p$.
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  • 857
9 votes

Covering a set with geometric progressions

We can reduce Lucia's upper bound of $3/8$ a little further as follows. Begin by taking the $n/4$ geometric progressions of common ratio $2$ beginning at each odd number at most $n/2$. Then for each ...
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9 votes
Accepted

Besicovitch Covering Lemma on Manifolds

In the Federer's book "Geometric Measure Theory", there is a notion of "directionally limited" metric space. He proves that the Besicovitch Covering Lemma holds for the directionally limited spaces. ...
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  • 1,737
8 votes
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induced group actions and covering maps on Eilenberg-Maclane space

If $M$ is connected, then @MarkGrant's fibration sequence gives a long exact sequence on homotopy groups showing that $\pi_1(M/\Sigma_k)\to \Sigma_k$ is surjective. Now apply the $K(-,1)$-functor and ...
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  • 666
8 votes
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Can any $n$ dimensional (smooth, PL, topological) closed manifold be covered by $2^n$ pieces of $n$ dimensional real spaces?

In the smooth, PL or topological category, you can cover a $n$-dimensional connected manifold by $n+1$ charts. Furthermore, for $k \leq n-3$ a $k$-connected $n$-dimensional manifold can be covered by ...
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  • 7,450
7 votes

Normal Covering of a Finite Group

I don't know references, but the following might be a starting point for an inductive approach. You might as well assume that each $N_{i}$ is maximal normal in $G.$ Now take a minimal normal subgroup $...
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7 votes

covering a square with unit squares

$\Pi(3)>10$ according to "A note on covering a square with equal squares", Januszweski, The American Mathematical Monthly Vol. 116, No. 2 (Feb., 2009), pp. 174-178.
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  • 171
7 votes

The minimal number of partitions to cover all $k$ tuples

Denote $N=\frac{2k\choose k}{2^k}$ and choose, say $m=\lceil 10kN\rceil$ independent random partitions (all partitions have equal probability $1/(2k-1)!!$). For any $k$-set $A$, the probability that ...
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  • 88.5k
6 votes

Easiest proof for showing finite etale (analytic) quotients of algebraic varieties are algebraic

Up to passing to the Galois closure $\bar{X} \to X$ and using Riemann Extension Theorem (ensuring that $\bar{X}$ is algebraic) we may assume that $X \to Y$ is a Galois cover, induced by the action of ...
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6 votes
Accepted

Minimum covers of complete graphs by $4$-cycles

If $n$ is odd, the answer is $\lceil \binom{n}{2}/4 \rceil$. If $n$ is even, the answer is $\lceil \binom{n}{2}/4+n/8 \rceil$. This follows from two special cases of a more general conjecture by ...
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  • 28.7k
4 votes

Normal Covering of a Finite Group

This is perhaps not quite on point, but I would like to mention a pretty little fact that is at least related to the question. If $G = \bigcup N_i$, where $N_i \triangleleft G$ and the $N_i$ are ...
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  • 5,979
4 votes
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Strongly minimal covers

Maybe I misunderstood something, but consider the following simple example. Let $V=(0,\infty)$ and let the maximal edges be the (open) unit intervals, except $(0,1)$. Any cover contains a sequence ...
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  • 17.2k
4 votes
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Inscribing a "chain" into an open cover

The answer is ``yes'' if $X$ is Hausdorff. We can identify the arc $C$ with the unit interval $[0,1]$ and assume that $[0,1]$ is contained in $X$ and is covered by a family $\mathcal U$ of open ...
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  • 33.6k
4 votes

Covering all except one of the purple intersection points of $n$ red and $m$ blue lines efficiently

Problem 6 from the 2007 IMO is related. That problem was to determine the least number of hyperplanes needed to cover $\{0,1,...,n\}^3 \setminus \{(0,0,0)\}$, and the answer is $3n$. In general, the ...
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  • 27.2k
4 votes
Accepted

Edge clique cover of a graph with restriction on how many times an edge can be covered

In general there can be a rather large difference in these quantities. If you take $k = 1$ you are considering the clique partition vs. clique covering problem. This is studied in Clique partitions ...
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4 votes
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Covering property of complete distributive lattices

I think you do mean completely distributive, not just finitely. Otherwise $\mathbb{Z}$ with its usual ordering is not a finite union of any set of closed intervals. For complete distributive lattices, ...
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4 votes
Accepted

Edge covers in infinite graphs

No. The proof requires a lemma: If $C$ is an edge cover with $|\mathrm{Good}(C)| < |E|$, then it is not minimal (I can remove a point without changing the fact that it's an edge cover). To prove ...
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  • 14.8k
4 votes
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Optimal pseudotransversals

No, here is an example of a hypergraph with no optimal transversal. Let $V=\omega$ (the set of nonnegative integers), and let $$E=\{\{0\}\} \cup\{\{0,n\};n\ge 1\} \cup \{\{i; i\ge n\}; n\in \omega\}....
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  • 5,436
4 votes
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Homology of universal abelian cover of a manifold

This has not much to do with $H_1(M_0)$. If $\pi :M_0\rightarrow M$ is your abelian covering, we have $\int_{\tau }\overline{\omega} =\int_{\pi _*\tau }\omega $. But the exact sequence $0\rightarrow \...
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  • 34.5k
4 votes
Accepted

Avoiding multiply covered vertices in graph edge coverings

For $n \in \mathbb N$ Let $a_n, b_n$ be a pair of vertices connected by an edge. For every finite subset $M \subset \mathbb N$ take an additional vertex $v_M$ and connect it to every vertex $a_n$ With ...
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4 votes
Accepted

Effect of snowflaking on doubling constants

Suppose $(X,d)$ has doubling constant $\lambda$. This means that for every $r$, every $d$-ball of radius $2r$ can be covered by $\lambda$ many $d$-balls of radius $r$. With respect to the metric $d^\...
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  • 37.4k
4 votes
Accepted

Is König's Property for graphs inheritable from finite subgraphs?

(Just making my comment an answer as suggested.) If every finite subgraph of $G$ satisfies Kőnig's Property, then $G$ has no odd cycles and is thus bipartite. Aharoni (König's Duality Theorem For ...
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  • 1,526
3 votes
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Covering all except one of the purple intersection points of $n$ red and $m$ blue lines efficiently

The very nice paper "Cayley-Bacharach theorems and conjectures" by David Eisenbud, Mark Green, and Joe Harris (http://www.ams.org/journals/bull/1996-33-03/S0273-0979-96-00666-0/home.html) gives an ...
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  • 4,735
3 votes

How to "lift" a transitive group action on a manifold?

The Palais theorem assumes that the manifold $\tilde M$ is compact. The positive answer gives proposition 6 of the Onishchik book "Topology of transitive transformation groups". It ...
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3 votes
Accepted

How to "lift" a transitive group action on a manifold?

If $G$ is connected, take its Lie algebra, acting as vector fields on $M$. Lift the vector fields by the covering map. There is a unique connected Lie group $\tilde{G}$ acting on $\tilde{M}$ whose Lie ...
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  • 23.4k
3 votes
Accepted

Finding a minimum covering of a polygon with interesting shapes

There is a recent paper by Tobias Christ that extends the covering work I cited in the comments, and work by Andrejz Lingas, to triangles, one of the problems left unresolved by that earlier work. ...
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3 votes

Minimal covering sets in families of sets intersecting in at most $1$ point

I think I can do one case. Assume that $A_i\subseteq S$, $|A_i|=\aleph_0$ for $i\in I$, and $|A_i\cap A_j|\leq 1$ for $i\neq j$. By a theorem in the paper P. Komjáth: Families close to disjoint ones,...
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