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Results tagged with big-list
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user 34180
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.
11
votes
Important formulas in combinatorics
The Kneser graph $KG_{n,k}$ is the graph on $k$-subsets of $\{1, \dots, n\}$ with two subsets made adjacent when they are disjoint. The formula $$\chi(KG_{n,k}) = n - 2k + 2$$ was proved by Lovász in …
10
votes
Most helpful math resources on the web
Sci-Hub is pretty helpful in accessing articles, even for those researchers who already have access to several journals. The interface is great, the site is pretty fast, and the database is huge. See …
10
votes
Proofs of the Chevalley-Warning Theorem
For Chevalley's theorem, i.e., number of common zeroes not being one, any new proof of the following Lemma would give a 'new' proof.
Lemma Let $P \in \mathbb{F}_q[x_1, \ldots, x_n]$ such that $P(a …
- 1,197
8
votes
Where have you used computer programming in your career as an (applied/pure) mathematician?
My PhD supervisor and I discovered a new near octagon, related to the finite simple group $G_2(4)$, using a computer. From this we were able to give a construction of the full Suzuki tower, which was …
7
votes
Not especially famous, long-open problems which anyone can understand
Assuming that the definitions of a graph, its diameter and girth are something anyone can understand*, whether a graph with diameter $2$, girth $5$ and degree $57$ exists or not is a long standing fam …
6
votes
Which math paper maximizes the ratio (importance)/(length)?
The 1949 paper by R.C. Bose "A Note on Fisher's Inequality for Balanced Incomplete Block Designs" arguably gave birth to the linear algebra method in combinatorics which has since been used by many to …
5
votes
Contest problems with connections to deeper mathematics
IMO 2007 P6. Let $n$ be a positive integer. Consider the set $S$ of points $(x, y, z)$ with $x, y, z \in \{0, 1, \dots, n\}$ and $x + y + z > 0$, so $S$ is a set of $(n+1)^3 - 1$ points in three-dimen …
5
votes
Combinatorial databases
Andries Brouwer's collection of strongly regular graphs: http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html
Eric Moorhouse's collections of finite projective planes and generalized polygons: http://eri …
4
votes
Linear algebra proofs in combinatorics?
Here are some examples where the dimension of a vector space of polynomials is used to solve a combinatorial problem.
Theorem 1 There are at most $n(n+1)/2$ equiangular lines in $\mathbb{R}^n$.
Proof. …
4
votes
Old books still used
"Projective Geometry" by Coxeter (1963), "Finite Geometries" by Dembowski (1968) and "Projective Planes" by Hughes and Piper (1973), still serve as great textbooks for these topics.