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Results tagged with big-list
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user 2233
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.
9
votes
Ingenuity in mathematics
Solving a math problem by appeal to electricity is pretty creative in my opinion. Check out squaring the square.
8
votes
What are some mathematical concepts that were (pretty much) created from scratch and do not ...
Although his work was certainly related to earlier fields, I believe that Ramanujan (pretty much) built up a lot of his work from scratch.
4
votes
Probabilistic method used to prove existence theorems
Regarding your second question, there is indeed a constructive proof showing that there are sparse graphs with arbitrarily high chromatic number due to Lovász (unfortunately behind a paywall). Howeve …
2
votes
What classification theorems have been improved by re-categorizing?
I (think) that the classification theorem of Chudnovsky, Robertson, Seymour and Thomas for Berge graphs qualifies. Namely, their strategy for proving the Strong perfect graph theorem was to prove the …
2
votes
17 camels trick
I think Strategy-stealing arguments from combinatorial game theory are in the same vein. Here is a classic example.
Prove that in two-move chess, Black does not have a winning strategy.
Proof. …
3
votes
Free open-access peer-reviewed math journals
Advances in Combinatorics is a new arXiv overlay journal that ''aims to be a top-level specialist journal in combinatorics.'' The journal has a very strong editorial board with Dan Král' and Tim Gower …
3
votes
Demonstrating that rigour is important
Richard Lipton recently blogged about this question in the context of why a potential proof of $P \neq NP$ would be important. I am probably bastardizing his words, but one of the reasons he gives is …
4
votes
Theorems for nothing (and the proofs for free)
I'd say the Tutte-Berge formula, which is a wonderful result that tells you (almost) everything you want to know about matchings in graphs. Although there are many proofs of this theorem, there is a …
9
votes
Which journals publish short notes in discrete mathematics?
Electronic Notes in Discrete Mathematics only publishes short notes in discrete mathematics.
28
votes
Most intricate and most beautiful structures in mathematics
The Turing degrees are an immensely intricate poset $\mathcal{D}$. Here are some of their remarkable properites:
Every countable poset is embeddable in $\mathcal{D}$.
$\mathcal{D}$ contains minimal …
22
votes
Generalizing a problem to make it easier
I like the matroid intersection theorem. Basically, this hammer often renders a problem trivial once you generalize to matroids. Also, it's nice that the hammer itself is not hard to prove. First t …
7
votes
When is 2 qualitatively different from 3?
The Banach-Tarski paradox holds in $\mathbb{R}^3$ but fails in $\mathbb{R}^2$.
2
votes
List of counting proofs instead of linear algebra method in combinatorics
Theorem. Let $A$ and $B$ be families of subsets of $[n]$, such that for all $a \in A$ and $b \in B$, $|a \cap b|$ is odd. Then $|A||B| \leq 2^{n-1}$.
I will present two proofs of this theorem. One …
- 32.1k
8
votes
Different ways of proving that two sets are equal
To expand on Pete's comment where we have additional structure, in any context where duality makes sense, to show that $A=B$, it suffices to show that they both have the same duals. Of course this do …
2
votes
What are the most fundamental classes of mathematical algorithms?
What about algorithms coming from graph theory? This seems to be a rich source with lots of real world applications too. To mention just a few
Minimum weight spanning tree
Maximum weight matchings …