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Results tagged with big-list
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user 8628
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
5
votes
Maximality statements that cannot be proved using $\mathsf{ZL}$
A statement I like very much is:
Any compact topology is contained in a maximal compact topology.
A direct application of Zorn's Lemma doesn't work, as the following argument shows. Cons …
- 48.9k
2
votes
Tweetable Mathematics
Erdös-Faber-Lovasz conjecture: If $n$ copies of $K_n$ have pairwise intersection of $\leq 1$, you can color all points with $n$ colors.
4
votes
Which mathematical definitions should be formalised in Lean?
Why formalize complex objects like topological groups when so much fun can be had with natural, simple objects like undirected graphs? There's also tons of open questions involving these simple beings …
3
votes
Prominent non-mathematical work of mathematicians
It has been noted already that Noam Elkies is an accomplished composer.
What I find at least as extraordinary about him is that he can hum-whistle some of Bach's two-part inventions. (Anecdotal "evide …
17
votes
Examples of common false beliefs in mathematics
Let $(X,\tau)$ be a topological space. The false belief is: "Every sequence $(x_n)$ in $X$ with an accumulation point $a\in X$ has a subsequence that converges to $a$". I subscribed to this intuitivel …
0
votes
Elementary + short + useful
I would show the students the Cantor-Schroeder-Bernstein Theorem and tell them they can themselves contribute to the list of elementary by providing a solution to the following open problems:
Does th …
20
votes
Examples of common false beliefs in mathematics
False belief: ${\cal P}(\omega)$ has only countable chains with respect to $\subseteq$.
It seems mind-boggling to me that you can start with $\emptyset$, and "add stuff" uncountably many …
5
votes
Solving algebraic problems with topology
Using Priestley duality for distributive lattices and compact, totally disconnected ordered topological spaces, many purely algebraic questions have been solved using quite simple topological tools. F …
1
vote
1
answer
359
views
Examples of "irregularities" in mathematics, other than prime numbers [closed]
Prime numbers are the prime example (no pun intended) for something that arises apparently without describable patters; we know that infinitely many exist, that gaps between them can be arbitrarily la …
4
votes
1
answer
321
views
Maximality without Zorn
When confronted with finding an object that is maximal with regard to some ordering relation, most of us have the reflex to use Zorn's Lemma.
I am interested in instances of proving the existence of …
- 48.9k
11
votes
Proposals for polymath projects
A conjecture that can be stated in so simple terms that it is hard to classify, is Frankl's Union-Closed Sets Conjecture. It would be fantastic to see this solved.
5
votes
Not especially famous, long-open problems which anyone can understand
If $2^x$ and $3^x$ are integers for some positive real number $x$, does this imply that $x\in\mathbb{N}$?
23
votes
Not especially famous, long-open problems which anyone can understand
The Graph Reconstruction Conjecture:
Let $G, H$ be finite, simple, loopless graphs such that $|V(G)|$ and $|V(H)|$ are at least $4$. If there is a bijection $\varphi:V(G)\to V(H)$ such that for all …
18
votes
Not especially famous, long-open problems which anyone can understand
The Littlewood conjecture:
For any $\alpha, \beta \in \mathbb{R}$ we have $$\lim\textrm{inf}_{n\to\infty} (n\cdot||n\alpha||\cdot||n\beta||) = 0$$
where $||\cdot||$ denotes the distance to the nea …
1
vote
What are examples of good toy models in mathematics?
Boolean algebras are toy models for distributive lattices, which in turn are toy models for lattices in general (and partially ordered sets).