1
vote
3answers
29 views

Prove: For any 2 coloring of 2-space, one of the color classes contains points at all distances

We color the 2D-plane either red or blue at every point. Prove that one of the sets (either red or blue), contains two points at distance $D$, for every positive real number $D$.
0
votes
2answers
42 views

Coloring of $K_{17}$

For any 3-coloring of $K_{17}$ I have to show there exists either a red, blue or green triangle. To start, can I use proof by contradiction with color red, blue, green? So $(0,0,136)$ means all 136 ...
0
votes
1answer
71 views

Two coloring questions and ramsays number

What is the smallest $n$ such that every 2-coloring of edges of $K_n$ contains a red or blue 4-cycle (not $K_4$)? I am given that $R(4,4) \le 18$ and $R(3,5) \le 14$ Any help is greatly appreciated!
4
votes
1answer
76 views

Euclidean Ramsey theory problem

Let $k\geq 1$ be given. Consider the following statement: For all (non equilateral) triangles (represented by 3 points in $\mathbb R^2$) and for all $k$-colorings of $\mathbb R^2$ there exists a ...
2
votes
2answers
138 views

What is a Ramsey Graph?

What is a ramsey graph and What is its relation to RamseyTheorem? In Ramsey Theorem: for a pairs of parameters (r,b) there exists an n such that for every (edge-)coloring of the complete graph on n ...
3
votes
1answer
136 views

example of a finite coloring without infinite monochromatic set closed under addition

I am studying some theorems on combinatorial set theory, especially Ramsey theorem and Hindman's theorem. I think I am going to ask a silly question, but I am too much involved in the subject to think ...
4
votes
2answers
138 views

Counterexample for $R(4,4) \neq 8$

I try to find a counterexample for $R(4,4)\neq 8$. (R is the Ramsey-number). I drew a graph with 8 vedges and I coloured all edges $(v_i,v_j)$ with $i-j =\pm 2,4,6$ in the same colour (for example ...
12
votes
1answer
309 views

Monochromatic squares in a colored plane

Color every point in the real plane using the colors blue,yellow only. It can be shown that there exists a rectangle that has all vertices with the same color. Is it possible to show that there exists ...
10
votes
1answer
97 views

Coloring of positive integers

Suppose $f:\mathbb{Z}^+\longrightarrow X$ is a function, with $X$ a finite set. Is it true that there are $a,b\in\mathbb{Z}^+$ such that $f(a)=f(b)=f(a+b)$.
2
votes
0answers
71 views

Monochromatic degenerate triangles in a two-coloring of the plane

In a similar vein to a question I asked a few days ago: Do all two-colorings of $\mathbb{R}^2$ contain three points of the same color which form the vertices of a degenerate triangle of side-lengths ...
3
votes
1answer
278 views

Gallai's theorem, colourings and equivalence relations

I'm revising a few past papers on Ramsey theory and I've come across a question which feels like it should be easy if it weren't so confusingly set up - I was hoping someone here could help me make ...
6
votes
1answer
163 views

Ramsey theory, finite colourings of $\mathbb{N}$ and infinite monochromatic sets

I am trying to show that the following statement is false: whenever $\mathbb{N}$ is finitely coloured by $c: \mathbb{N} \to \{1,\ldots,k\}$ (in the sense of Ramsey theory), there exists an infinite ...