# Tagged Questions

106 views

### (3n,n)-Turán graph [closed]

I'm working on a problem regarding (kn,n)-Turán graphs. The (2n,n)-Turán graph, also known as the cocktail party graph, has a closed formula for its number of spanning trees. I want to know if there ...
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### Colouring a chessboard

How can I demonstrate that I can colour a $2n\times\binom{2n}{2}$ chessboard, with $n$ different colours, such that there aren't $4$ separate unit squares of the same colour, the centers of which are ...
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### Coloring numbers from $1$ to $1000$

I mostly just need someone to explain to me this problem: Prove that it is possible to $2$-color the integers from $1$ to $1000$ so that no monochromatic arithmetic progression of length $17$ is ...
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### 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 ...
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### Triangle free graphs with large chromatic number

I am trying to understand the proof of Theorem 2 given here. (Page 5) The theorem states that $\forall k\exists$ a triangle free graph $G$ with $\chi(G)>k$. The proof constructs such a $G$ as ...
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### Planar graph with a chromatic number of 4 where all vertices have a degree of 4.

Is it possible to have a planar graph with a chromatic number of $4$ such that all vertices have degree $4$? Every time I try to make the degree condition to work on a graph, it loses its planarity.
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### Can a planar graph without two triangles that share an edge have a chromatic number larger than 3?

Let G be a square with one diagonal. Are there any planar graphs without G as a subgraph that are not 3-colourable?
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 $f(a)=f(b)=f(a+b)$.