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Results tagged with big-list
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user 1532
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.
3
votes
Ingenuity in mathematics
Let me mention that the question is closely related to a more recent one Proofs that require fundamentally new ways of thinking which I think also is about ingenuity in a sense. So indeed I was not su …
6
votes
Statements reliant on conjectures
In computational complexity there are several conjectures which are stronger than $NP \ne P$ which have important consequences. To mention a few
1) The conjecture that factoring is computationally h …
8
votes
Important formulas in combinatorics
$$\Theta (C_5)=\sqrt 5.$$
This is the formula by Lovasz for the Shannon capacity of the cycle of length 5.
The Shannon capacity of a graph $\Theta (G)= \lim_{n \to \infty}(\omega(G^n))^{1/n}$, where …
5
votes
Important formulas in combinatorics
The Kruskal-Katona inequality: $$|\partial{\cal F}| \ge {{m_k} \choose {k-1}} + {{m_{k-1}} \choose {k-2}}+ \cdots + {{m_j} \choose {j-1}}.$$
Here ${\cal F}$ is a family of $k$-sets, and $\partial …
6
votes
Tweetable Mathematics
Cap set problem solved: polynomial method, punchline going back to (a+b)²=a²+b²+2ab. link
18
votes
Tweetable Mathematics
Borsuk's conjecture - a counterexample for DIM>2000, the tensor product of the unit sphere with itself. Link
11
votes
What are some mathematical concepts that were (pretty much) created from scratch and do not ...
This is an intruiging question. I have some suggestions but I am not sure about them.
1) Frege's work on logic. (Logic was stagnated for many many centuries before.)
2) Conway's surreal numbers.
3) …
1
vote
Generalized notions of solutions in various areas of mathematics
One of the most fruitful notion of generalized solution in optimization and combinatorics is linear programming relaxation. Quoting from the wikipedia article: In mathematics, the linear programming r …
1
vote
Generalized notions of solutions in various areas of mathematics
A form of "generalized solution" which I saw in various areas like for combinatorial optimization problems, for diophanine equations, for computational complexity purposes, and others is "statistical …
4
votes
Important formulas in combinatorics
$$\frac {\alpha (G)}{n}\le \frac {\lambda_{\min}}{d-\lambda_{\min}}$$
This is Hoffman's bound for the independence number $\alpha (F)$ (namely, the largest number of vertices in an independent set of …
4
votes
Important formulas in combinatorics
$$(1-\lambda_2)/2\le h(G)\le \sqrt{2(1-\lambda_2)},$$ is the "discrete Cheeger-Buser inequality", relating the spectral gap of the discrete Laplace operator to the discrete Cheeger constant of a grap …
11
votes
What are the most overloaded words in mathematics?
The word "stable" is used in many different contexts. Also "elementary" has many usages. The word "lattice" has two entirely different meanings which are ar time confusing. So is the word "field".
I …
3
votes
Tweetable Mathematics
Lattices with exponential kissing number discovered by Serge Vlăduţ. Another home run for algebraic-geometry codes. Link.
Actually, with the new 260 characters policy we can add:
Lattices with expon …
14
votes
Tweetable Mathematics
This is actually a real tweet by Ryan O'Donnell on Huang's proof of the sensitivity conjecture.
Hao Huang@Emory:
Ex.1: ∃edge-signing of n-cube with 2^{n-1} eigs each of +/-sqrt(n)
Inte …
12
votes
Important formulas in combinatorics
$N({\cal A})= \sum _{x\in L({\cal A})}(-1)^{r(x)}\mu (0,x)$
This is Zaslavsky's formula for the number of regions in an arrangement of hyperplanes.
The details: Given an arrangement of hyperplanes $\c …