# Tagged Questions

Branch of combinatorics with the philosophy that 'total disorder is impossible'. For example, Ramsey's theorem asserts that for each $n$, every sufficiently large graph either contains a clique of size $n$ or a stable set of size $n$.

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### Van der Waerden like theorem

I am trying to develop bounds for the function B(k) where B(k) is defined as the least such positive integer so that whenever the set $\{1,2,\cdots B(k)\}$ is partitioned into two parts at least one ...
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### Hales Jewett Theorem

In the book Ramsey Theory by Graham, Rothschild and Spencer the authors state: The Hales Jewett Theorem strips van der Waerden's theorem of its unessential elements and reveals the heart of Ramsey ...
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### Large bicliques in r-partite graphs containing no independent sets having one vertex from each class

Let $G$ be a multipartite graph on $r$ classes, each containing $k$ vertices, such that there is no independent set which contains at least one vertex from each class. I believe such graphs contain a ...
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### The Problem about 2-coloring finite plane

Suppose we color a $X \times X$ finite plane by red and blue arbitrarily. How large does X need to be to guarantee a monochromatic combinatorial square $k \times k$ 1 0 1 0 1 1 1 1 1 1 1 1 1 ...
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### Quasi-Ramsey Graphs

A graph with number of vertices less than R(n,n)-1, is called quasi-Ramsey of case n, if it has no complete graph Kn in itself or its complementary graph, and if added another vertex, no matter how it ...
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### Density Ramsey theorems with explicit asymptotics

I wonder what interesting and non-trivial examples of density Ramsey theorems with explicit asymptotics are there? I'm aware of two examples: Szemerédi's theorem and density Hales-Jewett theorem. ...
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### Is this Ramsey-type problem an open problem?

A blog claims that the following Ramsey-type (or van der Waerden type) problem is open: If the natural numbers are colored with finitely many colors, must there exist x and y (not both 2) such that x+...
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### Constructive lower bounds for multicolor Ramsey numbers

The $k$-color Ramsey number of the complete graph $K_n$, denoted with $R_k(n)$, is defined to be the smallest integer $t$, such that in any $k$-coloring of the edges of $K_t$, there is a complete ...
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### 3x3 submatrix with only $0$ or $1$ entries

I decided to cross-post the question here from math.stackexchange.com because I got no answer from there. It is a quick question on bipartite Ramsey numbers (I'm not an expert on the subject, so ...
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### Arithmetic progressions of length 3 in subset of Z_n of size n^d

Let $A\subset\mathbb{Z}/n\mathbb{Z}$ such that: $|A|>n^{d}$ ($0< d <1$). Let $C=\{(x,y,2y-x)\in A\times A \times A\}$ be the set of $3$-term arithmetic progressions within $A$. [The ...
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### Cliques, Paley graphs and quadratic residues

A question I've thought about, on and off for a long time, is how to improve the best bounds that (seem to be) known for the clique numbers of Paley graphs. If p=1 mod 4 is a prime, we can define the ...
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### Best lower bound for off-diagonal Ramsey numbers

What are the current best lower bounds for off-diagonal Ramsey numbers $R(k,l)$ with $l$ of order unity and asking for asymptotic behavior for large $k$, such as $R(k,4)$, $R(k,5)$, and so on? (...
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### Ergodic limits along subsets of $\mathbb{N}.$

Let say that an infinite subsets $A$ of $\mathbb{N}$ is "nice w.r.to ergodic limits", if it can replace $\mathbb{N}$ in the individual ergodic theorem, that is, if it is such that the following ...
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### Differences of near diagonal Ramsey numbers.

I am a graduate student trying to get involved in Ramsey theory. My question comes from: Erdős on graphs: his legacy of unsolved problems By Fan R. K. Chung, Paul Erdős, Ronald L. Graham p.14 of ...
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### Any 2-coloring of the scope of equilateral triangle must create a monochromatic rectangular triangle ?

Hello All, I'm trying to prove something that we know that is correct but can't find how to do it or any example to show how to solve it. Any 2-coloring of the scope of equilateral triangle must ...
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### monochromatic cycle-free colouring of the complete graph on R?

Hi So there is an edge-colouring of a complete graph on R (the reals), with countably many colours that as no monochromatic triangle. To find it map R to (0,1) write the numbers in binary and if 2 ...
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### Monochromatic triangles in every two-coloring of the plane?

An old problem (possibly due to Erdős and Graham?): given a triangle $T$ and a two-coloring of the plane, does there necessary exist a monochromatic congruent copy of $T$? Here "monochromatic" means ...
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### Ramsey pairs of classes graphs

Motivation Call a set $H$ of vertices of a graph $G$ homogeneous if it is either independent or complete. Every perfect graph of size $n$ has a homogeneous subset of size $\sqrt n$. Noga Alon has ...
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### Proofs of Lower Bounds for Ramsey Numbers?

As a sort of dual question to this question, I am wondering what proofs people know of lower bounds on Ramsey numbers $R(k, k)$. I know of two proofs: there is Erdos's beautiful probabilistic ...
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### Noncombinatorial proofs of Ramsey's Theorem?

I know of 2(.5) proofs of Ramsey's theorem, which states (in its simplest form) that for all $k, l\in \mathbb{N}$ there exists an integer $R(k, l)$ with the following property: for any $n>R(k, l)$, ...
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### Ramsey multiplicity

Given a positive integer $a$, the Ramsey number $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K_n$ are colored using only two colors, we necessarily have a copy of $K_a$ ...
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### Suppose the independent number of a graph is bounded. How small the clique number can be?

Suppose the independent number of a graph is bounded. How small the clique number can be? linear? It seems to be a natural problem to ask. but I could not find any reference. Thanks.
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### Bounds on a partition theorem with ambivalent colors

I've been running into the following type of partition problem. Given positive integers h, r, k, and a real number ε ∈ (0,1), find n such that if every (unordered) r-tuple from an n ...
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### Erdos Conjecture on arithmetic progressions

Introduction: Let A be a subset of the naturals such that $\sum_{n\in A}\frac{1}{n}=\infty$. The Erdos Conjecture states that A must have arithmetic progressions of arbitrary length. Question: I ...
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### Where can I find a catalog of known Ramsey numbers?

Is there an online catalog available of Ramsey numbers, preferably one that for unknown values documents the known upper/lower bounds?
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### Applications of infinite Ramsey's Theorem (on N)?

Finite Ramsey's theorem is a very important combinatorial tool that is often used in mathematics. The infinite version of Ramsey's theorem (Ramsey's theorem for colorings of tuples of natural numbers) ...
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### Magnitude of Graham's Number?

I recently stumbled across this number, and then (foolishly, most likely) decided to try to describe it in a blog post http://frothygirlz.com/2010/01/14/big-numbers-part-2/ Q - Are there any ...
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### Ramsey Theory, monochromatic subgraphs

If we have the complete countably infinite bipartite graph $K_{\omega,\omega}$ and we colour the edges with just two colours. Should we expect to get a monochromatic copy of $K_{\omega,\omega}$. ...
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### Bound on cardinality of a union

Suppose I have n finite sets A1 through An contained in some fixed set S, and I am given non-negative integers N and N1 through Nn such that each Ai has cardinality N, and each k-tuple intersection ...
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### Non trivial colouring of the edges of an infinite complete graph

Can you build a probabilistic scheme for colouring each edge (independently of all other edges) of the complete graph G on the positive integers such that the probability that G contains an infinite ...
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### Infinite Ramsey theorem with infinitely many colours

Clearly, it is possible to colour the edges of an infinite complete graph so that it does not contain any infinite monochromatic complete subgraph. Now what about the following? Let $G$ be the ...
Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large ...