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

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

The first result we present holds definitionally and is useful for translating between results couched in typical Ramsey theory language and results given in standard graph theory language. Letting $G$ be any graph, we interpret it as a 2-coloring of the edges of $K_{|G|}$ by coloring all edges in $G$ red and the remaining edges (not present in $G$ ) blue. This coloring has no red $K_{\omega(G)+1}$ and no blue $K_{\alpha(G)+1}$ by definition. This yields the following result.

Lemma 3.15. Let $G$ be any graph. Then $R(\omega(G)+1, \alpha(G)+1) \geq|G|+1$.

Applying Lemma $3.15$ with $G$ being a 5 -cycle we easily have $\omega(G)=2$ and $\alpha(G)=2$ so that $R(3,3) \geq 6$, which as we have seen is tight.

The next general result is a special case of a result of Chvátal and Harary [46]. We remind the reader that all arbitrary graphs we consider are connected.
Theorem 3.16. Let $G$ and $H$ be graphs. Then
$$R(G, H) \geq(\chi(G)-1)(|H|-1)+1 .$$
Proof. Let
$$n=(\chi(G)-1)(|H|-1)$$
We will exhibit a 2-coloring of the edges of $K_{n}$ with no red $G$ and no blue $H$, from which the inequality follows. Partition the vertices of $K_{n}$ into $\chi(G)-1$ parts of $|H|-1$ vertices each. In each part, color all edges between the $|H|-1$ vertices blue. Color the remaining edges of $K_{n}$ red. Hence, every edge between copies of $K_{|H|-1}$ is red. Clearly this coloring admits no blue $H$ as $H$ cannot be a subgraph of $K_{|H|-1}$. We must show that there is no red $G$.

Assume, for a contradiction, that this coloring admits a red $G$. Then $G$ may have at most one vertex in each copy of $K_{|H|-1}$. Hence, $G$ can have at most $\chi(G)-1$ vertices. However, this means we can use $\chi(G)-1$ colors to color the vertices of $G$ to produce a vertex-valid coloring of $G$. This contradicts the definition of $\chi(G)$ as being the minimal such number.

Applying Theorem 3.16, we give our first result for specific types of graphs. This result is due to Chvátal [45]. Recall that $T_{m}$ is a tree on $m$ vertices.
Corollary 3.17. Let $m, n \in \mathbb{Z}^{+}$. For any given tree $T_{m}$, we have
$$R\left(T_{m}, K_{n}\right)=(m-1)(n-1)+1 .$$
Proof. Using $G=K_{n}$ and $H=T_{m}$ in Theorem $3.16$ we immediately have $R\left(T_{m}, K_{n}\right) \geq(m-1)(n-1)+1$ so it remains to show that $R\left(T_{m}, K_{n}\right) \leq$ $(m-1)(n-1)+1$. To show this we induct on $m+n$, with $R\left(T_{2}, K_{n}\right)=n$ and $R\left(T_{m}, K_{2}\right)=m$ being trivial. Note that the inductive assumption means the formula holds for any type of tree on less than $m$ vertices.
Let $K$ be the complete graph on vertices $V$ with
$$|V|=(m-1)(n-1)+1 .$$

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

Describing the die in Figure $3.5$ as a graph on 8 vertices, our edge set would be
$${{a, b},{a, d},{a, e},{b, c},{b, f},{c, d},{c, g},{d, h},{e, f},{e, h},{f, g},{g, h}} .$$
But this is not how we would normally describe a standard die. We typically note the 6 sides of a die and not the edges as we have given them. We may

call the sides faces; however, in order to abstract the idea of a graph, let’s call them some type of edge. For the same reason we abstract a plane in 3 dimensions to a hyperplane in more than 3 dimensions, we abstract edges of more than 2 vertices to hyperedges.

Recalling Definition 3.1, we may refer to the faces of our cube as hyperedges and describe the associated hypergraph as being on the vertices $a, b, c, d, e, f, g, h$ with hyperedge set
$${{a, b, c, d},{a, b, e, f},{a, d, e, h},{b, c, f, g},{c, d, g, h},{e, f, g, h}}$$
The reader may notice that each hyperedge in this description contains exactly 4 vertices. This is an example of a hypergraph in the class a hypergraphs with which we will be dealing.

Definition $3.19$ ( $\ell$-uniform hypergraph). Let $G=(V, E)$ be a hypergraph. Let $\ell \in \mathbb{Z}^{+}$with $\ell \geq 3$. If every element of $E$ contains exactly $\ell$ vertices from $V$, then we say that $G$ is an $\ell$-uniform hypergraph. If $E$ contains all ( $\ell \mid$ subsets of $V$ of size $\ell$, then we call $G$ the complete $\ell$-uniform hyperyraph and denote it by $K_{n}^{\ell}$, where $n=|V|$.

Applying this definition to the die example above, our second description is one of a 4-uniform hypergraph; however, it is not a complete 4-uniform hypergraph since it has only 6 hyperedges and not all $\left(\begin{array}{l}6 \ 4\end{array}\right)=15$ hyperedges that a $K_{6}^{4}$ has.

## 数学代写|离散数学作业代写discrete mathematics代考|Hypergraph Ramsey Theorem

In Ramsey’s Theorem we color edges and deduce monochromatic complete subgraphs. Replacing edges with hyperedges and subgraphs with subhypergraphs is the basis of the Hypergraph Ramsey Theorem, which is actually the original theorem proved by Ramsey in his seminal paper [166, Theorem A].
Formally, for a hypergraph $H=(V, E)$, each hyperedge in $E$ is assigned a color and a subhypergraph on vertices $U \subseteq V$ may only have a hyperedge $e \in E$ provided $e \in \wp(U)$.

Notation. We will refer to a hyperedge consisting of $\ell$ vertices as an $\ell$ hyperedge.

Theorem 3.20 (Hypergraph Ramsey Theorem). Let $\ell, r \in \mathbb{Z}^{+}$with both at least 2. For $i \in{1,2, \ldots, r}$, let $k_{i} \in \mathbb{Z}^{+}$with $k_{i} \geq \ell$. Then there exists a minimal positive integer $n=R_{\ell}\left(k_{1}, k_{2}, \ldots, k_{r}\right)$ such that every $r$-coloring of the $\ell$-hyperedges of $K_{n}^{\ell}$ with the colors $1,2, \ldots, r$ contains, for some $j \in$ ${1,2, \ldots, r}$, a $K_{k_{j}}^{\ell}$ subhypergraph with all $\ell$-hyperedges of color $j$.

Proof. We start by proving the $r=2$ case. We prove this via induction on $\ell$. The $\ell=2$ case of this theorem is Ramsey’s Theorem (Theorem $3.6$ ), so that we have already proved the base case. Given $k_{1}$ and $k_{2}$, by the inductive assumption we assume that $n=R_{\ell-1}\left(k_{1}, k_{2}\right)$ exists. For ease of exposition, we

will use the colors red and blue, with red associated with $k_{1}$ (and blue with $\left.k_{2}\right)$.

Inside of the induction on $\ell$, we induct on $k_{1}+k_{2}$, as we did in the proof of Ramsey’s Theorem. So, in our pursuit of showing that $R_{\ell}\left(k_{1}, k_{2}\right)$ exists, we may assume that $R_{\ell}\left(k_{1}-1, k_{2}\right)$ and $R_{\ell}\left(k_{1}, k_{2}-1\right)$ both exist. The base cases for our induction on $k_{1}+k_{2}$ are trivial: $R_{\ell}\left(\ell, k_{2}\right)=k_{2}$ and $R_{\ell}\left(k_{1}, \ell\right)=k_{1}$ (for the first, we either have a red hyperedge and, hence, a red $K_{\ell}^{\ell}$, or all hyperedges are blue and we have a blue $K_{k_{2}}^{\ell}$; for the second, reverse the colors and use $k_{1}$ instead of $k_{2}$ ). Hence, we may assume that $k_{1}$ and $k_{2}$ are both greater than $\ell$. Let
$$n=R_{\ell-1}\left(R_{\ell}\left(k_{1}-1, k_{2}\right), R_{\ell}\left(k_{1}, k_{2}-1\right)\right)+1 .$$
We will show that $R_{\ell}\left(k_{1}, k_{2}\right) \leq n$.
Consider an arbitrary 2 -coloring $\chi$ of the $\ell$-hyperedges of $K_{n}^{\ell}$ with vertex set $V$. Isolate a vertex $v \in V$. Consider the complete $(\ell-1)$-uniform hypergraph on vertices $V \backslash{v}$ where the coloring $\widehat{\chi}$ of the $(\ell-1)$-hyperedges is inherited from $\chi$ in the following way: For $e$ an $(\ell-1)$-hyperedge, let
$$\widehat{\chi}(e)=\chi(e \cup{v}) .$$
From the definition of $n$ we have either a complete $(\ell-1)$-uniform hypergraph on $R_{\ell}\left(k_{1}-1, k_{2}\right)$ vertices with all $(\ell-1)$-hyperedges red under $\hat{\chi}$ or a complete $(\ell-1)$-uniform hypergraph on $R_{\ell}\left(k_{1}, k_{2}-1\right)$ vertices with all $(\ell-1)$-hyperedges blue under $\hat{\chi}$. Without loss of generality, we may assume the latter holds.

Let $W$ be the set of $R_{\ell}\left(k_{1}, k_{2}-1\right)$ vertices with all $(\ell-1)$-hyperedges blue under $\hat{\chi}$. By the assumed existence of $R_{\ell}\left(k_{1}, k_{2}-1\right)$ we have, under $\chi$, either a red $K_{k_{1}}^{\ell}$ and are done, or we have a blue $K_{k_{2}-1}^{\ell}$ on vertex set $U$ with the property that all $(\ell-1)$ hyperedges are blue under $\widehat{\chi}$. Noting that $v \notin U$, consider the complete $\ell$-uniform hypergraph on the vertices $U \cup{v}$. The $\ell$ hyperedges that include $v$ are all blue since, by the definition of $\widehat{\chi}$, we have $\widehat{\chi}(e)=\chi(e \cup{v})$ for any $(\ell-1)$-hyperedge $e$ with vertices in $V \backslash{v}$, which we have deduced are all blue in $K_{k_{2}-1}^{\ell}$. Thus, all $\ell$-hyperedges on $U \cup{v}$ are blue and we have a blue $K_{k,}^{\ell}$. This completes the $r=2$ case of the theorem.

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

R(G,H)≥(χ(G)−1)(|H|−1)+1.

n=(χ(G)−1)(|H|−1)

R(吨米,ķn)=(米−1)(n−1)+1.

|在|=(米−1)(n−1)+1.

## 数学代写|离散数学作业代写discrete mathematics代考|Hypergraph Ramsey Theorem

n=Rℓ−1(Rℓ(ķ1−1,ķ2),Rℓ(ķ1,ķ2−1))+1.

χ^(和)=χ(和∪在).

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