2
votes
0answers
92 views

reference on aperiodicity and cluster [closed]

From this image: I believe there is a message relating those clusters drawn in picture and aperiodic tiling. Does anyone have some reference on this? Thank you :)
0
votes
0answers
90 views

topological space of Wang Tile

When trying to reprove a theorem in Wang tile: An established proof in Wang Tile which I doubt , a few notions are provided which I would like to seek for more information: For a given set of blocks ...
3
votes
0answers
119 views

characterization of all periodic tiling of a simple set of Wang Tile

Consider a set of Wang Tile such that all the edges are either 1 or 0.... there are 16 elements in such a set. Now, I wish to characterize all the periodic tilings of this set (better if they are ...
14
votes
1answer
1k views

What have simplicial complexes ever done for graph theory?

(I am asking in a somewhat tongue-in-cheek fashion, of course, but nevertheless...) Are there examples of results in "classical" [*] graph theory that have been achieved by using simplicial ...
4
votes
4answers
386 views

Determine if a graph has a large clique

This question is quite specific and practical. I hope it is still relevant for MO and will not be removed. I have a collection $\mathcal{C}$ of graphs having from 5000-6000 vertices and edge density ...
4
votes
1answer
183 views

An intutive reason why a “distance” metric may be a poor one for a procedure where we attempt to modify a string (mutating 0 OR 1 bits)

If I'm attempting to mutate one arbitrarily chosen binary string $s_a$, to another arbitrarily chosen binary string $s_b$, in the smallest number of steps (i.e. with the smallest number of mutations) ...
4
votes
1answer
447 views

a conjecture in sum-free sets

Let $ A $ be a set of non-zero integers. Then $A$ contains a sum-free subset $B$ of size $ |B|> \frac{|A|}{3} $ (a result of Erdős). It is conjectured that RHS can be improved to $\frac{|A|}{3} ...
2
votes
5answers
252 views

Equivalence relations not associated with a group

This is a vague question; so vague that I wonder if anyone will get it. Many, perhaps most, equivalence relations that are regularly used in mathematics correspond to the orbits of some group action ...
7
votes
2answers
2k views

Two questions about combinatorics journals

Hello, I have two questions regarding combinatorics journals. I hope that this is the right place for such questions. Which combinatorics/DM journals would you consider as the "top tier"? I tried ...
23
votes
9answers
2k views

Is the empty graph a tree?

This is a boring, technical question that I stumbled upon while making a contribution to Sage. I would still like to hear a constructive answer so hopefully the question does not get closed. The ...
6
votes
4answers
863 views

fourier analytic proofs

While searching through Mathoverflow, I found out a fourier analytic proof of the Isoperimetric Inequality.Also, by google search I found a fourier analytic proof of Quadratic Reciprocity theorem.I ...
7
votes
3answers
508 views

Intuition Behind a Decimal Representation with Catalan Numbers

From $0 = 0.5 - 0.5 = 0.5 - \sqrt{0.25}$, we can adjust the subtrahend slightly to obtain $$0.5 - \sqrt{0.249} = 0.001\ 001\ 002\ 005\ 014\ 042\ldots$$ where the decimal representation contains the ...
15
votes
0answers
862 views

Origins of the Nerve Theorem

Recently, I've read two papers which have cited the Nerve Theorem, one crediting Borsuk with the result and another Leray. Here is the question: Who was the first to prove the Nerve Theorem?
17
votes
6answers
2k views

True by accident (and therefore not amenable to proof)

The graph reconstruction conjecture claims that (barring trivial examples) a graph on n vertices is determined (up to isomorphism) by its collection of (n-1)-vertex induced subgraphs (again up to ...
3
votes
2answers
551 views

Open problems and known identities involving sums

As many people here, I know of a few identities involving expressions of the type $\sum_{i}\ f(i)$, with "arbitrarily complicated $f(\cdot)$", as well as closed formulas in some cases. I also know ...
9
votes
4answers
2k views

A learning roadmap for Additive combinatorics.

Hello, I'd love to learn more about the field of additive combinatorics. From what I've understand, there's a book by Tao and Vu out on the subject, and it looks fun, but I think I lack the ...
17
votes
4answers
703 views

Arrow's theorem and the postseason

There are a number of instances of sports teams intentionally losing matches in order to secure a more favorable situation in a playoff round. While this doesn't happen terribly often, when it does ...
9
votes
2answers
628 views

Advantageous properties of 4 letter alphabet (in DNA)?

As you know DNA is composed of strings of 4 letters. I am wondering if the number 4 here has any significance? Any property of 4 that makes using 4 letters more advantageous over more (or less) ...
5
votes
3answers
256 views

Medium-Sized Calculations and Organization

This is not a math question as much as a process question. For the first time in my (very short) career, I find myself doing one of those messy calculations, where each 'line' of the calculation can ...
8
votes
4answers
1k views

the delta system lemma outside set theory

The lemma: Any uncountable set $S$ of finite sets has an uncountable subset $\Delta \subseteq S$ and an $x$ such that $\forall a,b \in \Delta$, if $a \neq b$ then $a \cap b = x$. $\Delta$ is called a ...
3
votes
9answers
1k views

Are there any important mathematical concepts without discrete analog?

In "continuous" mathematics there are several important notions such as covering space, fibre bundle, Morse theory, simplicial complex, differential equation, real numbers, real projective plane, etc. ...
31
votes
5answers
2k views

Why do wedges of spheres often appear in combinatorics?

Robin Forman writes in "A User's Guide to Discrete Morse Theory": The reader should not get the impression that the homotopy type of a CW complex is determined by the number of cells of each ...
14
votes
11answers
6k views

Different ways of proving that two sets are equal

I'm not sure if this is a soft question, or should be community wiki. I was explaining to a student how to prove that two sets were equal using what I called the 'oldest trick in the book': to show ...
4
votes
1answer
1k views

Combinatorics journals processing time

This is a spin-off question from How to select a journal?. Is there is any data available regarding processing time (acceptance time, time from submission to publication, or similar) specifically for ...
23
votes
6answers
6k views

Submitting to arXiv when unaffiliated

I am writing a short paper in the area of combinatorics. When the paper is complete, I would like to be able to submit it to arXiv. The reasons that I would like to submit to arXiv are: To obtain ...
5
votes
2answers
1k views

Yet another graph invariant: the similarity matrix

Preliminaries Let $n \in \mathbb{N}$ and $v$ be a vertex of a graph $G$. Let the $n$-neighbourhood of $v$, $N_n(v)$, be the induced subgraph of $G$ containing $v$ and all vertices at most $n$ edges ...
11
votes
8answers
33k views

The factorial of -1, -2, -3, …

Well, n! is for integer n < 0 not defined -- as yet. So the question is: How could a sensible generalization of the factorial for negative integers look like? Clearly a good generalization should ...
9
votes
10answers
808 views

Algorithmic Combinatorics resources?

Some branches of combinatorics lend themselves naturally to algorithms; graph theory is a natural example. However, straight-up enumerative combinatorics relies much more on analytic and algebraic ...
8
votes
3answers
529 views

“Plateaus” to watch out for

I'm a lot earlier in my math education that most of the people on this site. Currently I'm studying computer science, and I'm interested in looking into statistical and optimization applications, as ...