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9
votes
2answers
1k views

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 $n>R(k, l)$, ...
18
votes
2answers
721 views

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 $K_a$ ...
2
votes
1answer
376 views

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.
5
votes
2answers
248 views

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 ...
21
votes
5answers
4k views

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 ...
3
votes
2answers
505 views

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?
15
votes
10answers
3k views

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) ...
12
votes
6answers
15k views

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 ...
8
votes
2answers
894 views

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 $K_{\omega,\omega}$. ...
5
votes
5answers
967 views

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 ...
5
votes
1answer
314 views

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 ...
5
votes
2answers
642 views

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 ...
12
votes
4answers
755 views

Splitting pythagorean triples

Can one partition the set of positive integers into finitely many pythagorean-triple-free subsets? If so, what is the smallest number of such subsets? Taking a wild guess, I would be least surprised ...
8
votes
5answers
946 views

Can one make Erdős's Ramsey lower bound explicit?

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 ...