### 数学代写|离散数学作业代写discrete mathematics代考|Graph Ramsey Theory

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## 数学代写|离散数学作业代写discrete mathematics代考|Complete Graphs

We start by reminding the reader of a few definitions about graphs.
Definition 3.1 (Graph, Hypergraph, Degree, Edge, Hyperedge). Let $V$ be a set, called the vertices, and let $E$ be a subset of $\wp(V) \backslash \emptyset$ (the power set of $V$ excluding the empty set). We say that $G=(V, E)$ is a graph if all elements of $E$ are subsets of size 2. In this case, $E$ is referred to as the set of edges. For any given vertex $v$, the number of edges containing $v$ is called the degree of

$v$. We say that $G=(V, E)$ is a hypergraph if some element of $E$ is a subset of more than 2 elements. In this case, $E$ is called the set of hyperedges.

Definition $3.2$ (Subgraph, Subhypergraph). Let $G=(V, E)$ be a graph. If $V^{\prime} \subseteq V$ and $E^{\prime} \subseteq E$ then $H=\left(V^{\prime}, E^{\prime}\right)$ is a subgraph of $G$. If $G$ is a hypergraph, then we call $H$ a subhypergraph.

We will start by considering graphs (hypergraphs are treated later in this chapter). We will assume that any arbitrary graph we consider is connected so that there exists a string of edges from any vertex to any other vertex. We also assume that our graphs are simple, meaning that at most one edge exists between any two vertices. We will start by considering one of the fundamental graphs: the complete graph.

Definition $3.3$ (Complete graph). Let $n \in \mathbb{Z}^{+}$. The complete graph on $n$ vertices, denoted $K_{n}$, is $G=(V, E)$ where $|V|=n$ and $E$ consists of all subsets of $V$ of size 2 .

We typically identify the vertex set $V$ of $K_{n}$ with ${1,2, \ldots, n}$ and the edge set $E$ with ${{i, j}: i, j \in[1, n] ; i<j}$. In other words, all $\left(\begin{array}{c}n \ 2\end{array}\right)$ possible edges are present in a complete graph.

We start with the classical version of Ramsey’s Theorem, restricted to the 2-color situation.

Theorem 3.4. Let $k, \ell \in \mathbb{Z}^{+}$. There exists a minimal positive integer $n=$ $n(k, \ell)$ such that if each edge of $K_{n}$ is assigned one of two colors, say red and blue, then $K_{n}$ admits a complete subgraph on $k$ vertices with all edges red or a complete subgraph on $\ell$ vertices with all edges blue.

Before getting to the proof, let’s consider the $k=\ell=3$ case. We will refer to $K_{3}$ as a triangle and will show that $n(3,3) \leq 6$. Consider any 2-coloring of the edges of $K_{6}$. Isolate one vertex; call it $X$. Then $X$ is connected to the other 5 vertices with either a red or a blue edge. Let $R$ be the set of vertices connected to $X$ with a red edge; let $B$ be the set of vertices connected to $X$ with a blue edge. We know that $|R \cup B|=5$ and that $R \cap B=\emptyset$. Hence, $|R \sqcup B|=|R|+|B|=5$. From this we can conclude by the pigeonhole principle that either $|R| \geq 3$ or $|B| \geq 3$. Without loss of generality, we assume $|R| \geq 3$.
At this stage we have $X$ connected to at least 3 vertices, call them $A, B$, and $C$, via red edges. Consider the edges between these latter 3 vertices. If any edge is red, say ${A, B}$, then we have a red triangle (on vertices $A, B$, and $X)$. In other words, we have deduced a $K_{3}$ with all edges red. On the other hand, if no edge is red, then they are all blue and we can conclude that the triangle on vertices $A, B$, and $C$ has all blue edges, i.e., we have a $K_{3}$ with all blue edges.

The proof for general $k$ and $\ell$ follows the same idea as the case for $k=\ell=$ 3 . We will isolate one vertex and separate the other vertices according to the color of the edge to the isolated vertex.

## 数学代写|离散数学作业代写discrete mathematics代考|Deducing Schur’s Theorem

As our first “application” of Ramsey’s Theorem, we will deduce Schur’s Theorem. Recall that Schur’s Theorem states that there exists a minimal positive integer $s(r)$ such that every $r$-coloring of $[1, s(r)]$ admits a monochromatic solution to $x+y=z$. So, we need to deduce integer solutions from a graph. We do so by considering a subclass of colorings of $K_{n}$.

Definition 3.7 (Difference coloring). A difference coloring of the edges of $K_{n}$ is one in which the color of every edge ${i, j}$ depends solely on $|i-j|$.

With this definition, if $\chi:[1, n-1] \rightarrow{0,1, \ldots, r-1}$ is a coloring of integers, then we have the induced difference coloring of $K_{n}$ where we color edge ${i, j}$ with $\chi(|i-j|)$.

To deduce Schur’s Theorem, let $n=R(3 ; r)$. For any $r$-coloring of $[1, n-1]$ consider the difference coloring of the edges of $K_{n}$. By Ramsey’s Theorem, this difference coloring admits a monochromatic $K_{3}$, say on the vertices $u, v$, and $w$, with $u<v<w$. This means that the integers $v-u, w-v$, and $w-u$ all have the same color. Let $x=v-u, y=w-v$, and $z=w-u$ to see that we have a solution to $x+y=z$ with $x, y$, and $z$ all the same color. Consequently, we see that $s(r) \leq R(3 ; r)-1$.

The above argument can be easily extended to show that any $r$-coloring of the integer interval [1, R( $k ; r)-1]$ admits a monochromatic solution to $\sum_{i=1}^{k-1} x_{i}=x_{k}$. The minimal positive integer $n=n(k ; r)$ such that every $r-$ coloring of $[1, n]$ admits a monochromatic solution to $\sum_{i=1}^{k-1} x_{i}=x_{k}$ is referred to as a generalized Schur number. This argument gives $n(k ; r) \leq R(k ; r)-1$. It is known [21] that
$$n(k ; 2)=k^{2}-k-1$$
however, $n(k ; r)$ for $r \geq 3$ is unknown: the bound
$$n(k ; r) \geq \frac{k^{r+1}-2 k^{r}+1}{k-1}$$has been determined [21]. As we can see, the bound between these two Ramsey-type numbers is quite weak since we have seen that $(\sqrt{2})^{k}<R(k, k)$ (see Inequality (3.1)) so that the $k^{2}-k$ lower bound on $R(k, k)$ is not strong. The reason for this is because we do not use the whole complete monochromatic subgraph in deducing the existence of $n(k ; r)$.

## 数学代写|离散数学作业代写discrete mathematics代考|Other Graphs

As stated at the end of the last section, the proof that generalized Schur numbers exist as a consequence of Ramsey’s Theorem does not use the full power of Ramsey’s Theorem. In particular, we do not need a monochromatic complete graph; we only need the “outside” edges of this complete graph to have the same color, i.e., a monochromatic cycle.

Definition $3.8$ (Path, Cycle, Tree). Let $G=(V, E)$ be the graph with vertex set $V=\left{v_{1}, v_{2}, \ldots, v_{n}\right}$ and edge set $E=\left{\left{v_{i}, v_{i+1}\right}: i \in{1,2, \ldots, n-1}\right}$. Then $G$ is called a path (from $v_{1}$ to $v_{n}$, or $v_{n}$ to $v_{1}$ ) and we denote it by $P_{n}$. If we add the edge $\left{v_{1}, v_{n}\right}$ to the edge set, then $G$ is called an $n$-cycle and we denote it by $C_{n}$. Paths are a subclass of the class of graphs known as trees. A tree $T_{n}$ is a graph on $n$ vertices with no $k$-cycle as a subgraph, for any $k$.
There are many other named graphs (for example, in Figure $3.2$ we present a graph attributed to Kempe but commonly referred to as the Peterson graph. which is useful as it often serves as a counterexample). Complete graphs, paths, cycles, and trees are some of the important ones. Below we define two more important classes of graphs.

## 数学代写|离散数学作业代写discrete mathematics代考|Deducing Schur’s Theorem

n(ķ;2)=ķ2−ķ−1

n(ķ;r)≥ķr+1−2ķr+1ķ−1已确定[21]。正如我们所看到的，这两个 Ramsey 型数之间的界限非常弱，因为我们已经看到(2)ķ<R(ķ,ķ)（见不等式（3.1）），因此ķ2−ķ下限R(ķ,ķ)不强。这是因为我们没有使用整个完全单色子图来推导n(ķ;r).

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## MATLAB代写

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