# Questions tagged [counterexamples]

A counterexample is an example that disproves a mathematical conjecture or a purported theorem. For example, the Peterson graph is a counterexample to many seemingly plausible conjectures in Graph Theory.

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### Are isomorphic maximal tori stably conjugate?

Let $F$ be a field and $G$ a reductive $F$-group. For various applications it is important to understand the "classes" of maximal ($F$-)tori of $G$. Here "class" can mean the ...
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### If chromatic polynomials for two graphs agree, can I always find an edge such that the two deletion-contraction minors have same chromatic polynomial?

Suppose I have non-isomorphic graphs $G$ and $H$ (which have at least one edge), but such that their chromatic polynomials are the same. Can I then always find an edge $e$ in $G$ and $f$ in $H$ such ...
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### Example of a pseudomonad on Cat whose pseudoalgebras are not the pseudoalgebras for a 2-monad

For every pseudomonad $T$ on the 2-category of (locally small) categories $\mathbf{Cat}$, we can consider the 2-category of $T$-pseudoalgebras and pseudomorphisms $T\text{-PsAlg}_p$, which is equipped ...
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### Pairwise orthogonality for partitions of unity in a *-algebra

Let $\mathcal{A}$ be a $*$-algebra with unit $1_{\mathcal{A}}$. As in the $\mathrm{C}^*$-setting, a projection is an element $p\in\mathcal{A}$ such that $p=p^2=p^*$. A partition of unity is a finite ...
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### Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points

The following question resisted attacks at Math SE, so I thought I would try posting it here. Is the following conjecture true or false: Given any five coplanar points, we can always draw at least ...
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### Isometric embeddings of metric $K_{n+1}$ in $\mathbb{R}^n$

Question: is it always possible to embed a complete, symmetric and metric graph $G$ with $n+1$ vertices isometrically in $\mathbb{R}^n$? I'm convinced it must be true, but can't remember having seen ...
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1 vote
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### Are the categories of definable dinatural transformations freely generated from discrete graphs?

It is well known that the dinatural transformations between multivariant functors defined in Functorial polymorphism don't form a category, because they do not compose in general, but some do. For any ...
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### An example of radical ideal which is irreducible but not prime

$\DeclareMathOperator\rad{rad}$I am searching an example of ideal $I$ of a ring $R$ such that $\rad(I)$ is irreducible but not prime ideal. In case $R$ is Noetherian, the radical of $I$ being ...
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### Cut a homotopy in two via a "frontier"

Consider a space $G$ obtained by glueing two disjoint cobordisms (the fact that they are might be irrelevant, assume they are topological spaces at first) $L$ and $R$ on a common boundary $C$. (...
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### Does $\mathbf{Cat}$ have the Cantor–Schröder–Bernstein property?

I am wondering if the category of small categories $\mathbf{Cat}$ is known to (not) have the Cantor–Schröder–Bernstein property? That is, for any two categories $\mathcal{C}$ and $\mathcal{D}$, does ...
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### Composition of power series is power series?

$\DeclareMathOperator\dom{dom}$Sorry to bother the community again with these type of questions about power series, I am ready to delete the question if it is not suitable. Definition: I say a ...
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### Local equality of functions implies global equality?

The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k_i\}_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. ...
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### Examples of isomorphic non-equivalent twisted group algebras

Let $F$ be a field, $G$ be a finite group and $\alpha \in Z^2(G, F^*)$ . The twisted group algebra $F^{\alpha}G$ is a $F$-algebra with $F$ vector basis, $\{\bar g : g \in G \},$ and multiplication ...
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### Lie algebras for which all one-dimensional extensions split

I was recently trying to prove the following "well-known" theorem for myself, given that I could not find a proof in the literature that I could understand. In what that follows, I will ...
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### When does the refinement of a paracompact topology remain paracompact?

Let $(X,\tau)$ be a Hausdorff paracompact space. Let $\tau'$ be the smallest $P$-topology refining $(X,\tau)$, i.e. the topology which has for base the $G_\delta$-subsets of $(X,\tau)$. Is it true ...
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### Example of a compact operator that is not uniformly continuous

I want to find a Banach space $E$ and a compact operator $K:[0,1]\times E \rightarrow E$ (that is, $K$ maps every bounded sequence onto a sequence that converges up to a subsequence) satisfying the ...
1 vote
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### What can be said about cluster sets for power series of two variables?

I'm still trying to prove the continuity of a function $u$ which can be interpreted as the restriction of a power series of two variables, which I haven't managed to approach the right way yet. To ...
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### Does the difference of solutions of two unrelated PDE solve an 'intermediate' equation?

I should preface this question by saying that I strongly suspect the answer is negative, but I couldn't find the counterexample myself. Say we are working on the unit disc $D \subset \mathbf{R}^n$, ...
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