# Tagged Questions

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### Using Ramsey theory to show some properties of subgraph of a directed graph

Let $r > 1$ be an integer. Prove that there is an integer $n_0$ such that for every integer $n\geq n_0$ and every directed graph $G = (\{1,...,n\}, E)$ without loops, $G$ has an induced ...
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### Ramsey Number for Star graphs

For two graphs $H_1$ and H2, the Ramsey number $r(H_1, H_2)$ is the minimum number r so that in any red-blue coloring of the edges of the complete graph Kr on r vertices there is necessarily either a ...
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### Schur's theorem and infinite version

I've got a homework exercise on Schur's theorem, which says that for any $r \in \mathbb N$ there is an $n \in \mathbb N$ such that for any $r$-colouring of $[n] := \{1, \dots, n\}$ there is a ...
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### Ramsey's theorem [closed]

I'm reading introduction to combinatorics and encountered an exercise I couldn't answer Let S be a set of six points in the plane, with no three of the points collinear. Color either red or blue each ...
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### Upper bound for ramsey number $r(a_1,\ldots, a_m)$

I am looking for any (finite) upper bound of the ramsey number $r(a_1,\ldots, a_m)$. I can prove the well known fact for any positive integers $a,b$ there is a $c$ for which $c\ge r(a,b)$ by taking ...
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### Ramsey number for paths

Let $n = R(P_{r+1}, c)$ be the smallest integer such that if $K_n$ is $c$-edge-coloured, then it contains a monochromatic subgraph isomorphic to $P_{r+1}$, the path of length $r$. I need to show that ...
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### 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 ...
### Lower bound for monochromatic triangles in $K_n$
Say $K_n$ is a complete graph of $n$ nodes, and every edge is either blue or red. I'm trying to find $T_n$, which is the lower bound for the number of monochromatic triangles in $K_n$ (monochromatic ...
### Prove that $r(k,k) + k \leq r(k + 1, k + 1)$
Prove that $r(k,k) + k \leq r(k + 1, k + 1)$, where $r(k,l)$ is the minimum number of vertexes in a Graph, where we have a clique with $k$ vertexes or a stable set with $l$ vertexes. There are ...