Ramsey Chains in Linear Forests

Abstract

Every red–blue coloring of the edges of a graph G results in a sequence $G_1$, $G_2$, …, $G_{\mathsf{\ell }}$ of pairwise edge-disjoint monochromatic subgraphs $G_i$ ($1\le i\le \mathsf{\ell }$) of size i, such that $G_i$ is isomorphic to a subgraph of $G_{i+1}$ for $1\le i\le \mathsf{\ell }-1$. Such a sequence is called a Ramsey chain in G, and $A{R}_c\left(G\right)$ is the maximum length of a Ramsey chain in G, with respect to a red–blue coloring c. The Ramsey index $AR\left(G\right)$ of G is the minimum value of $A{R}_c\left(G\right)$ among all the red–blue colorings c of G. If G has size m, then $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$ for some positive integer k. It has been shown that there are infinite classes S of graphs, such that for every graph G of size m in S, $AR\left(G\right)=k$ if and only if $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$. Two of these classes are the matchings $m{K}_2$ and paths $P_{m+1}$ of size m. These are both subclasses of linear forests (a forest of which each of the components is a path). It is shown that if F is any linear forest of size m with $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)<m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$, then $AR\left(F\right)=k$. Furthermore, if F is a linear forest of size $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$, where $k\ge 4$, that has at most $\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)$ components, then $AR\left(F\right)=k$, while for each integer t with $\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)<t<\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ there is a linear forest F of size $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ with t components, such that $AR\left(F\right)=k-1$.

1. Introduction

For every graph G of size m, there is a unique positive integer k, such that $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$. The graph G is said to have an ascending subgraph decomposition {$G_1$, $G_2$, …, $G_k$} into k (pairwise edge-disjoint) subgraphs of G if $G_i$ is isomorphic to a proper subgraph of $G_{i+1}$ for $i=1,2,\dots ,k-1$. The following conjecture was stated in [1].

The Ascending Subgraph Decomposition Conjecture Every graph has an ascending subgraph decomposition.

This conjecture has drawn the attention of many researchers but has never been proved or disproved. There are many papers dealing with this conjecture (see [2,3,4,5,6,7,8,9], for example). Among the several classes of graphs for which the conjecture has been verified are regular graphs (see [3,10]). Two classes of graphs for which the conjecture can easily be verified are matchings $m{K}_2$ (consisting of m components $K_2$) and stars $K_{1,m}$: that is, for every positive integer m, where $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$, there is an ascending subgraph decomposition {$G_1$, $G_2$, …, $G_k$} of the graph G if either $G=m{K}_2$ or $G=K_{1,m}$, such that $G_i$ is isomorphic to a proper subgraph of $G_{i+1}$ for $1\le i\le k-1$. If $G=m{K}_2$, the subgraphs $G_i$ are matchings and if $G=K_{1,m}$, each subgraph $G_i$ is a star.

By a red–blue edge coloring (or simply a red–blue coloring) of a graph G, every edge of G is colored red or blue. Such an edge coloring is also referred to as a 2-edge coloring. In [11], the concept of ascending subgraph decomposition was extended to graphs possessing a red–blue coloring. Suppose that a red–blue coloring of a graph $G=m{K}_2$ or $G=K_{1,m}$ is given, where $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$. It was shown in [12] that there is not only an ascending subgraph decomposition {$G_1$, $G_2$, …, $G_k$} of G but one in which each subgraph is monochromatic as well. This (perhaps unexpected) observation led to another concept, which is related to some topics in Ramsey Theory, named for the British mathematician Frank Ramsey [13]. Ramsey theory is one of the most studied areas in combinatorics and graph theory, with many highly nontrivial and beautiful results (see [14,15,16,17,18,19,20,21,22], for example). We refer to the books [11,23] for basic definitions and notation in graph theory that are not defined here.

Let G be a graph (without isolated vertices) of size m with a red–blue coloring c. A subgraph H of G is monochromatic if all edges of H are colored the same. A Ramsey chain of G with respect to c is a sequence $G_1$, $G_2$, …, $G_{\mathsf{\ell }}$ of pairwise edge-disjoint subgraphs of G, such that each subgraph $G_i$ ($1\le i\le \mathsf{\ell }$) is monochromatic of size i and $G_i$ is isomorphic to a subgraph of $G_{i+1}$ for $1\le i\le \mathsf{\ell }-1$. Each subgraph $G_i$ ($1\le i\le \mathsf{\ell }$) in a Ramsey chain is called a link of the chain. The maximum length of a Ramsey chain of G with respect to c is the (ascending) Ramsey index $A{R}_c\left(G\right)$ of G. The Ramsey index $AR\left(G\right)$ of G is defined by $AR\left(G\right)=min\{A{R}_c\left(G\right):c\mathrm{is}\mathrm{a}\mathrm{red}–\mathrm{blue}\mathrm{edge}\mathrm{coloring}\mathrm{of}G\}.$ Consequently, if $AR\left(G\right)=k$ for some graph G, then for every red–blue coloring of G, there is a Ramsey chain of length k in G, while there exists at least one red–blue coloring for which there is no Ramsey chain of length greater than k. These concepts were introduced and studied in [11,12], using somewhat different technology, and they were studied further in [24,25]. An immediate observation on Ramsey indexes of graphs was presented in [12].

Observation 1 

([12]). If G is a graph of size m where $2\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$ for a positive integer k, then $AR\left(G\right)\le k$.

The result obtained on matchings and stars can therefore be stated as follows:

Theorem 1 

([12]). If $G\in \{m{K}_2,K_{1,m}\}$, where $m\ge 3$, then

$$AR(G)=kifandonlyif\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right).$$

In [24], the question was posed as to whether there are other infinite classes S of graphs, such that for every sufficiently large integer m and each graph G of size m in S, it follows that $AR\left(G\right)=k$ if and only if $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$. In [24,25], this concept was studied for cycles and paths. As the emphasis here is on the size of a graph, let $C_m$ denote a cycle of size m and $Q_m$ a path of size m: that is, $Q_m$ is a path of order $m+1$.

Theorem 2 

([12,25]). If $G\in \{C_m,Q_m\}$, where $m\ge 3$, then

$$AR(G)=kifandonlyif\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right).$$

A linear forest is a forest of which every component is a path. Here, we are only concerned with linear forests without isolated vertices. Paths and matchings are linear forests, namely, the linear forests with the minimum and maximum number of components. The goal here is to determine whether Theorems 1 and 2 can be extended to include linear forests distinct from paths and matchings.

2. Ramsey Indexes of Linear Forests

We saw in Theorems 1 and 2 that if $G=m{K}_2$ or $G=Q_m$ for a positive integer m, then $AR\left(G\right)=k$ if and only if $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$. As $m{K}_2$ and $Q_m$ are both linear forests, this raises the question whether the same result holds for all linear forests of size m. This question can be answered immediately. The linear forest $F=Q_2+(m-2)K_2$ consisting of $m-1$ components where $m=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ with $k\ge 3$ has Ramsey index $k-1$, not k. The red–blue coloring of F that assigns red to both edges of $Q_2$ and blue to all other edges does not result in a Ramsey chain of length k, since a Ramsey chain $F_1,F_2,\dots ,F_k$ of length k would require that $F_2=Q_2$ and $F_3=3K_2$; however, $F_2⊈F_3$. On the other hand, $\left(\genfrac{}{}{0pt}{}{k}{2}\right)K_2\subseteq F$ and $AR\left(\left(\genfrac{}{}{0pt}{}{k}{2}\right)K_2\right)=k-1$ by Theorem 1; therefore, $AR\left(F\right)=k-1$ by Observation 2.

Each of the linear forests $Q_2+8K_2$, $Q_3+7K_2$, $Q_4+6K_2$, $2Q_3+4K_2$, $Q_4+Q_3+3K_2$, $Q_7+3K_2$ has size $10=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$, where $k=4$. In each of these linear forests, a red–blue coloring is given in Figure 1 that shows that its Ramsey index is $3=k-1$. In Figure 1, a bold edge indicates a red edge and a thin edge indicates a blue edge.

Each of the six linear forests in Figure 1 has t components for $t=4,5,\dots ,9$. The examples in Figure 1 suggest that determining $AR\left(F\right)$ for a linear forest F may depend not only on its size but the number of components of F as well. First, we present a result that gives the Ramsey index of linear forests of size m, where $m\ne \left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ for any positive integer k. Prior to doing this, we state some useful information from three results presented in [24,25].

Observation 2 

([24]). If H and G are graphs, such that $H\subseteq G$, then $AR\left(H\right)\le AR\left(G\right)$. Consequently, if $AR\left(H\right)\ge k$ for each graph H of size m, then $AR\left(G\right)\ge k$ for every graph G of size $m+1$.

Theorem 3 

([25]). Let $n\ge 5$ be an integer. For every set $\{n_1,n_2,\dots ,n_t\}$ of t integers, such that $1\le n_1<n_2<\dots <n_t\le \lceil n/2\rceil$ and ${\sum }_{i=1}^t{n}_i=n$, every linear forest of size n can be decomposed into the matchings $n_1{K}_2$, $n_2{K}_2$, …, $n_t{K}_2$.

Proposition 1 

([25]). Let $m=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ for some integer $k\ge 5$. For every two positive integers $m_1$ and $m_2$ with $m=m_1+m_2$ and $m_1,m_2\notin \{2,4\}$, there exists a partition of $\left[k\right]=\{1,2,\dots ,k\}$ into two subsets $A=\{a_1,a_2,\dots ,a_{k_1}\}$ and $B=\{b_1,b_2,\dots ,b_{k_2}\}$, where $k_1+k_2=k$, $a_1<a_2<\dots <a_{k_1}\le ⌈\frac{m_1}{2}⌉$, and $b_1<b_2<\dots <b_{k_2}\le ⌈\frac{m_2}{2}⌉$, such that ${\sum }_{i=1}^{k_1}{a}_i=m_1$ and ${\sum }_{i=1}^{k_2}{b}_i=m_2$.

We now apply Observations 1 and 2, Theorem 3, and Proposition 1, to establish the following result.

Theorem 4. 

If H is a linear forest of size m, where $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)<m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$ for some integer $k\ge 3$, then $AR\left(H\right)=k$.

Proof. 

As $AR\left(H\right)\le k$ by Observation 1, it remains only to verify that $AR\left(H\right)\ge k$: that is, to verify that H has a Ramsey chain of length k for every 2-edge coloring of H. It suffices to assume that $m=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)+1$ by Observation 2. Let c be a 2-edge coloring of H using the colors 1 and 2. We show that there is a Ramsey chain of length k in H. For $i=1,2$, let $H_i$ be the linear forest in H induced by the edges of H colored i. Let $H_i$ have size $m_i$ where $1\le m_1\le m_2$. Therefore, $m_1+m_2=m=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)+1$. We now consider five cases, according to whether $m_1\in \{1,2,3,4\}$ or $m_1\ge 5$.

Case 1: $m_1=1$; thus, $m_2=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$. As $k\ge 3$, it follows that $k\le ⌈\frac{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)}{2}⌉$. Because $1+2+\dots +k=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$, it follows by Theorem 3 that $H_2$ can be decomposed into $K_2,2K_2,3K_2,\dots ,k{K}_2$, which is a Ramsey chain of length k in H.

Case 2: $m_1=2$; thus, $m_2=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-1$. As $k\ge 3$, it follows that $k\le ⌈\frac{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-1}{2}⌉$. Because $2+3+\dots +k=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-1$, it follows by Theorem 3 that $H_2$ can be decomposed into $2K_2,3K_2,\dots ,k{K}_2$. As $K_2\subseteq H_1$, there is a Ramsey chain of length k in H.

Case 3: $m_1=3$; thus, $m_2=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2$. Suppose first that $k=3$; then, $m_2=4$. The linear forest $H_1$ can be decomposed into $K_2$ and $2K_2$, while the linear forest $H_2$ contains either $3K_2$ or $Q_2+K_2$. In either case, H contains a Ramsey chain of length $k=3$. Hence, we may assume that $k\ge 4$. Let F be a linear forest of size $m^{\prime }=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-3$ in $H_2$. As $k\ge 4$, it follows that $k\le ⌈\frac{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-3}{2}⌉$. Because $3+4+\dots +k=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-3$, it follows by Theorem 3 that F can be decomposed into $3K_2,4K_4,\dots ,k{K}_2$. As $H_1$ can be decomposed into $K_2$ and $2K_2$, there is a Ramsey chain of length k in H.

Case 4: $m_1=4$; thus, $m_2=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-3\ge 7$, and so, $k\ge 4$. Therefore, $k\le ⌈\frac{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-3}{2}⌉$. As $3+4+\dots +k=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-3$, it follows by Theorem 3 that $H_2$ can be decomposed into $3K_2,4K_4,\dots ,k{K}_2$. Let F be a linear forest of size 3 in $H_1$. As F can be decomposed into $K_2$ and $2K_2$, there is a Ramsey chain of length k in H.

Case 5: $m_1\ge 5$; thus, $m_2=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)+1-m_1$ and $k\ge 5$. Let F be a linear forest of size $m_2^{\prime }=m_2-1\ge 5$ in $H_2$. Consequently, $m_1\notin \{2,4\}$, $m_2^{\prime }\notin \{2,4\}$, and $m_1+m_2^{\prime }=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$. By Proposition 1, there exists a partition of $\left[k\right]=\{1,2,\dots ,k\}$ into two subsets $A=\{a_1,a_2,\dots ,a_{k_1}\}$ and $B=\{b_1,b_2,\dots ,b_{k_2}\}$, where $k_1+k_2=k$, $a_1<a_2<\dots <a_{k_1}\le ⌈\frac{m_1}{2}⌉$ and $b_1<b_2<\dots <b_{k_2}\le ⌈\frac{m_2^{\prime }}{2}⌉$, such that ${\sum }_{i=1}^{k_1}{a}_i=m_1$ and ${\sum }_{i=1}^{k_2}{b}_i=m_2^{\prime }$. Hence, $H_1$ can be decomposed into $a_1{K}_2,a_2{K}_2,\dots a_{k_1}{K}_2$ and F can be decomposed into $b_1{K}_2,b_2{K}_2,\dots b_{k_2}{K}_2$, resulting in a Ramsey chain $K_2,2K_2,\dots ,k{K}_2$ of length k in H. □

We illustrate the proof of Theorem 4 for the linear forest $Q_5+Q_6$ of size $m=11$ consisting of paths of sizes 5 and 6. Then, $k=4$ and $\left(\genfrac{}{}{0pt}{}{4+1}{2}\right)<11<15=\left(\genfrac{}{}{0pt}{}{4+2}{2}\right)$. Five red–blue colorings of $Q_5+Q_6$ are given in Figure 2, to reflect the five cases for $m_1=1,2,3,4,5$ in the proof, where a bold edge indicates a red edge and a thin edge indicates a blue edge. For $i=1,2,3,4$, an edge labeled i belongs to the link $G_i$ in a Ramsey chain $R:G_1,G_2,G_3,G_4$ of length 4 in $Q_5+Q_6$, and the unlabeled edge is not used in R.

3. Binomial Linear Forests

From the results obtained in Section 2, it follows that investigating $AR\left(F\right)$ for a linear forest F of size m, we need only be concerned when $m=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ for some integer $k\ge 3$. Thus, it is convenient to introduce terminology for this class of linear forests. A linear forest is k-binomial (or simply binomial) if its size is $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ for some positive integer k. For example, a 3-binomial linear forest has size 6, a 4-binomial linear forest has size 10, and a 5-binomial linear forest has size 15. We begin with an observation that is a consequence of Theorem 4 and Observation 2.

Corollary 1. 

If H is a k-binomial linear forest where $k\ge 3$, then $AR\left(H\right)\in \{k-1,k\}$. Furthermore, both values $k-1$ and k are attainable.

Proof. 

First, $AR\left(H\right)\le k$ by Observation 1. By Theorem 4, every linear forest of size $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-1$ has Ramsey index $k-1$. Hence, $AR\left(H\right)\ge k-1$ by Observation 2. Therefore, $AR\left(H\right)\in \{k-1,k\}$. As $AR(Q_2+(\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2)K_2)=k-1$ and $AR(Q_{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)})=k$, both values $k-1$ and k are attainable. □

By Theorem 4 and Corollary 1, if H is a 3-binomial linear forest, then $AR\left(H\right)\in \{2,3\}$. We now determine the exact value of the Ramsey index of every 3-binomial linear forest. All 3-binomial linear forests are listed below:

$$\begin{gathered} Q_6,Q_5+K_2,Q_4+Q_2,Q_4+2K_2,2Q_3,Q_3+Q_2+K_2, \\ {Q}_3+3K_2,3Q_2,2Q_2+2K_2,Q_2+4K_2,6K_2. \end{gathered}$$

Proposition 2. 

Let H be a 3-binomial linear forest. Then, $AR\left(H\right)=2$ if and only if

$$H\in \{Q_4+2K_2,Q_3+3K_2,Q_2+4K_2\}.$$

Proof. 

Let $X=\{Q_4+2K_2,Q_3+3K_2,Q_2+4K_2\}$. First, observe that the linear forests H in X are the only 3-binomial linear forests containing a subgraph $F=Q_2$, such that $H-E\left(F\right)=4K_2$ and, consequently, contains no copies of $Q_2$. Let c be a 2-edge coloring of H that assigns the color 1 to the two edges of F and the color 2 to all other edges of H. We claim that there is no Ramsey chain of length 3 in H. Assume, to the contrary, that there is a Ramsey chain $G_1,G_2,G_3$ of length 3 in H. As the size of H is 6, it follows that $\{G_1,G_2,G_3\}$ is a decomposition of H. Necessarily, $G_2=F=Q_2$, and so, $Q_2\subseteq G_3$. As $H-E\left(F\right)=4K_2$, this is impossible. Therefore, $AR\left(H\right)=2$ by Corollary 1.

For the converse, suppose that $H\notin X$. We show that $AR\left(H\right)=3$. As $AR\left(H\right)\le 3$ by Observation 1, it suffices to show that $AR\left(H\right)\ge 3$. Let c be a 2-edge coloring of H using the colors 1 and 2, where $m_i$ is the number of the edges in the subgraph $H_i$ of H colored i for $i=1,2$ and $m_1\le m_2$. Thus, $1\le m_1\le 3$ and $m_1+m_2=6$. We consider all possible pairs $(m_1,m_2)$:

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If $(m_1,m_2)=(1,5)$, then $G_1=H_1=K_2$ and $H_2$ is a linear forest of size 5. As $2+3=5$, it follows by Theorem 3 that $H_2$ can be decomposed into $2K_2,3K_2$. Thus, $K_2,2K_2,3K_2$ is a Ramsey chain of length 3 in H.

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If $(m_1,m_2)=(2,4)$, then $G_2=H_1\in \{Q_2,2K_2\}$. First, suppose that $H_1=2K_2$. Then, $H_2$ is a linear forest of size 4, and so, $H_2$ can be decomposed into $K_2$ and a subgraph $G_3$ of size 3. As every graph of size 3 contains $2K_2$, it follows that $K_2,H_1=2K_2,G_3$ is a Ramsey chain of length 3 in H. Next, suppose that $H_1=Q_2$. As $H\notin X$, it follows that $H_2\ne 4K_2$. Thus, $1\le k\left(H_2\right)\le 3$. If $k\left(H_2\right)=1$, then $H_2=Q_4$ can be decomposed into $K_2$ and $Q_3$. Thus, $K_2,Q_2,Q_3$ is a Ramsey chain of length 3 in H. If $k\left(H_2\right)=2$, then $H_2=2Q_2$ or $H_2=Q_3+K_2$. As $H_2$ can be decomposed into $K_2$ and $Q_2+K_2$, it follows that $K_2,Q_2,Q_2+K_2$ is a Ramsey chain of length 3 in H. If $k\left(H_2\right)=3$, then $H_2=Q_2+2K_2$. As $H_2$ can be decomposed into $K_2$ and $Q_2+K_2$, it follows that $K_2,Q_2,Q_2+K_2$ is a Ramsey chain of length 3 in H.

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If $(m_1,m_2)=(3,3)$, then $H_1$ can be decomposed into $K_2$ and $2K_2$ and $H_2$ contains $2K_2$. Thus, $K_2,2K_2,H_2$ is a Ramsey chain of length 3 in H.

□

We now consider k-binomial linear forests for integers $k\ge 4$. The following two results provide useful information on 2-edge colorings of k-binomial linear forests for $k\ge 4$.

Proposition 3. 

For a k-binomial linear forest H where $k\ge 4$, let c be a 2-edge coloring of H using the colors 1 and 2. For $i=1,2$, let $H_i$ be the subgraph of H induced by the edges of H colored i, where $H_i$ has size $m_i$ and $1\le m_1\le m_2$. If $H_1\notin \{Q_2,2Q_2,Q_4\}$, then $K_2,2K_2,\dots k{K}_2$ is a Ramsey chain of length k in H and $A{R}_c\left(H\right)=k$.

Proof. 

We consider five cases, depending on whether $m_1\in \{1,2,3,4\}$ or $m_1\ge 5$.

Case 1: $m_1=1$. Let $H_1=K_2$. As $k\le ⌊\frac{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-1}{2}⌋$ for $k\ge 4$ and $2+3+\dots +k=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-1$, it follows by Theorem 3 that $H_2$ can be decomposed into $2K_2,3K_2,\dots ,k{K}_2$. Thus, $K_2,2K_2,\dots k{K}_2$ is a Ramsey chain of length k in H and $A{R}_c\left(H\right)=k$.

Case 2: $m_1=2$. Thus, $H_1\in \{2K_2,Q_2\}$ and $m_2=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2$. As $H_1\ne Q_2$, it follows that $H_1=2K_2$. Because $H_2$ is a linear forest of size $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2$ and $1+3+4\dots +k=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2$ and $k\le ⌊\frac{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2}{2}⌋$ when $k\ge 4$, it follows by Theorem 3 that $H_2$ can be decomposed into $K_2,3K_2,4K_2,\dots ,k{K}_2$; thus, $K_2,2K_2,3K_2,\dots ,k{K}_2$ is a Ramsey chain of length k in H and $A{R}_c\left(H\right)=k$.

Case 3: $m_1=3$. The subgraph $H_1$ can be decomposed into $K_2$ and $2K_2$. As $k\le ⌊\frac{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-3}{2}⌋$ for $k\ge 4$ and $3+\dots +k=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-3$, it follows by Theorem 3 that $H_2$ can be decomposed into $3K_2,4K_2,\dots ,k{K}_2$. Thus, $K_2,2K_2,\dots k{K}_2$ is a Ramsey chain of length k in H and $A{R}_c\left(H\right)=k$.

Case 4: $m_1=4$. Then, $H_1\in \{Q_4,Q_3+K_2,2Q_2,Q_2+2K_2,4K_2\}$ and $m_2=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-4$. As $H_1\notin \{2Q_2,Q_4\}$, it follows that $H_1\in \{Q_2+2K_2,Q_3+K_2,4K_2\}$:

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Let $k=4$. Then, $H_2$ is a linear forest of size 6. As $1+2+3=6$, it follows by Theorem 3 that $H_2$ can be decomposed into $K_2,2K_2,3K_2$. Thus, $K_2,2K_2,3K_2,H_1$ is a Ramsey chain of length 4 in H.

★

Let $k\ge 5$. Then, $H_1$ can be decomposed into $K_2$ and $3K_2$. As $2+4+5+\dots +k=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-4$ and $k\le ⌊\frac{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-4}{2}⌋$ when $k\ge 5$, it follows by Theorem 3 that $H_2$ can be decomposed into $2K_2,4K_2,5K_2,\dots ,k{K}_2$. Thus, $K_2,2K_2,\dots ,k{K}_2$ is a Ramsey chain of length k in H and $A{R}_c\left(H\right)=k$.

Case 5: $m_1\ge 5$. By Proposition 1, there exists a partition of $\left[k\right]=\{1,2,\dots ,k\}$ into two sets $A=\{a_1,a_2,\dots ,a_{k_1}\}$ and $B=\{b_1,b_2,\dots ,b_{k_2}\}$, where $k_1+k_2=k$, $a_1<a_2<\dots <a_{k_1}\le ⌈\frac{m_1}{2}⌉$ and $b_1<b_2<\dots <b_{k_2}\le ⌈\frac{m_2}{2}⌉$, such that ${\sum }_{i=1}^{k_1}{a}_i=m_1$ and ${\sum }_{i=1}^{k_2}{b}_i=m_2$. As $m_2\ge m_1\ge 5$, it follows by Theorem 3 that $H_1$ can be decomposed into the matchings $a_1{K}_2$, $a_2{K}_2$, …, $a_{k_1}{K}_2$ and $H_2$ can be decomposed into the matchings $b_1{K}_2$, $b_2{K}_2$, …, $b_{k_2}{K}_2$. Consequently, $K_2,2K_2,\dots k{K}_2$ is a Ramsey chain of length k in H and $A{R}_c\left(H\right)=k$. □

Proposition 4. 

For a k-binomial linear forest H where $k\ge 4$, let c be a 2-edge coloring of H using the colors 1 and 2. For $i=1,2$, let $H_i$ be the subgraph of H induced by the edges of H colored i, where $H_i$ has size $m_i$ and $1\le m_1\le m_2$. If

(a) 

$H_1=Q_2$ and $H_2$ has at least $k-2$ pairwise edge-disjoint copies of $Q_2$ or

(b) 

$H_1\in \{Q_4,2Q_2\}$ and $H_2$ has at least $k-3$ pairwise edge-disjoint copies of $Q_2$,

then $Q_1$, $G_2\in \{Q_2,2Q_1\}$, $Q_2+Q_1$, $Q_2+2Q_1$, …, $Q_2+(k-2)Q_1$ is a Ramsey chain of length k in H.

Proof. 

First, suppose that $H_1=Q_2$ and $H_2$ has at least $k-2$ pairwise edge-disjoint copies of $Q_2$, which are denoted by $A_3$, $A_4$, …, $A_k$. Then, the number of edges of $H_2$ not belonging to any $A_i$ ($3\le i\le k$) is

$$\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2-2(k-2)=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2k+2=\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+1,$$

and so, these $\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+1$ edges of $H_2$ can be decomposed into

$$B_1=Q_1,B_3=Q_1,B_4=2Q_1,B_5=3Q_1,\dots ,B_k=(k-2)Q_1,$$

in such a way that $A_i+B_i=Q_2+(i-2)Q_1$ for $3\le i\le k$. Consequently, $Q_1$, $Q_2$, $Q_2+Q_1$, $Q_2+2Q_1$, …, $Q_2+(k-2)Q_1$ is a Ramsey chain of length k in H.

Next, suppose that $H_1\in \{Q_4,2Q_2\}$. Then, $H_1$ can be decomposed into $G_1=Q_1$ and $G_3=Q_2+Q_1$. The subgraph $H_2$ has at least $k-3$ pairwise edge-disjoint copies of $Q_2$, which are denoted by $A_4$, $A_5$, …, $A_k$. Then, the number of edges of $H_2$ not belonging to any $A_i$ ($4\le i\le k$) is

$$\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-4-2(k-3)=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2k+2=\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+1,$$

and so, these $\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+1$ edges of $H_2$ can be decomposed into

$$B_2=2Q_1,B_4=2Q_1,B_5=3Q_1,\dots ,B_k=(k-2)Q_1,$$

in such a way that $A_i+B_i=Q_2+(i-2)Q_1$ for $4\le i\le k$. Consequently, $Q_1$, $2Q_1$, $Q_2+Q_1$, $Q_2+2Q_1$, …, $Q_2+(k-2)Q_1$ is a Ramsey chain of length k in H. □

With the aid of Observation 1 and Propositions 3 and 4, we are able to establish the following result.

Theorem 5. 

If H is a k-binomial linear forest, where $k\ge 4$ with at most $\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)$ components, then $AR\left(H\right)=k$.

Proof. 

As $AR\left(H\right)\le k$ by Observation 1, it remains to show that $AR\left(H\right)\ge k$. Thus, we show that every 2-edge coloring of H produces a Ramsey chain of length k in H. Let c be a 2-edge coloring of H using the colors 1 and 2. For $i=1,2$, let $H_i$ be the linear forest in H induced by the edges of H colored i, where $H_i$ has size $m_i$ with $1\le m_1\le m_2$ and $k\left(H_i\right)$ components. By Proposition 3, if $H_1\notin \{Q_2,2Q_2,Q_4\}$, then there is a Ramsey chain of length k in H. Thus, we may assume that $H_1\in \{Q_2,2Q_2,Q_4\}$. We consider these two cases, depending on whether $H_1=Q_2$ or $H_1\in \{Q_4,2Q_2\}$.

Case 1: $H_1=Q_2$. Then, $H_2$ is a linear forest of size $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2$. Let $k\left(H_2\right)=\mathsf{\ell }$. Then, $\mathsf{\ell }\le k\left(H\right)+1\le \left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+1$. Let $J_1,J_2,\dots ,J_{\mathsf{\ell }}$ be the components of $H_2$, where $J_i$ has size $q_i$ for $1\le i\le \mathsf{\ell }$. Thus, ${\sum }_{i=1}^{\mathsf{\ell }}{q}_i=m_2=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2$. Let p be the maximum number of pairwise edge-disjoint copies of $Q_2$ in $H_2$. In each component $J_i$ ($1\le i\le \mathsf{\ell }$), the maximum number of pairwise edge-disjoint copies of $Q_2$ is $\lfloor q_i/2\rfloor$, and so, at most one edge of $J_i$ does not belong to these $\lfloor q_i/2\rfloor$ pairwise edge-disjoint copies of $Q_2$ in $J_i$. Hence, at most one edge in each $J_i$ ($1\le i\le \mathsf{\ell }$) does not belong to any of these p pairwise edge-disjoint copies of $Q_2$ in $H_2$. As $\mathsf{\ell }\le \left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+1$, it follows that

$$p\ge \frac{1}{2}\left[\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2-\mathsf{\ell }\right]\ge \frac{1}{2}\left[\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2-\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)-1\right]=k-2.$$

Therefore, there are at least $k-2$ pairwise edge-disjoint copies of $Q_2$ in $H_2$. By Proposition 4, there is a Ramsey chain of length k in H.

Case 2: $H_1\in \{Q_4,2Q_2\}$. Then, $H_1$ can be decomposed into $G_1=Q_1$ and $G_3=Q_2+Q_1$. Here, $H_2$ is a linear forest of size $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-4$. Let $k\left(H_2\right)=\mathsf{\ell }$. Then, $\mathsf{\ell }\le k\left(H\right)+2\le \left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+2$. Let $J_1,J_2,\dots ,J_{\mathsf{\ell }}$ be the components of $H_2$, where $J_i$ has size $q_i$ for $1\le i\le \mathsf{\ell }$. Thus, ${\sum }_{i=1}^{\mathsf{\ell }}{q}_i=m_2=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-4$. Let ${\mathsf{\ell }}^{\prime }$ be the number of these components having odd size. If $J_i$ has even size, then let $J_i^{\prime }=J_i$. If $J_i$ has odd size, then let $J_i^{\prime }$ be the subgraph of $J_i$ obtained by removing a pendant edge from $J_i$, where $J_i^{\prime }$ is empty if $q_i=1$. Hence, every subgraph $J_i^{\prime }$ has even size $q_i^{\prime }$ for $1\le i\le \mathsf{\ell }$ and every nonempty linear forest $J_i^{\prime }$ can be decomposed into $\frac{q_i^{\prime }}{2}$ copies of $Q_2$. The size of the linear forest $H_2^{\prime }$ consisting of $J_1^{\prime },J_2^{\prime },\dots ,J_{\mathsf{\ell }}^{\prime }$ is, therefore, ${\sum }_{i=1}^{\mathsf{\ell }}{q}_i^{\prime }=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-4-{\mathsf{\ell }}^{\prime }$, which is an even number. As $\mathsf{\ell }\le k\left(H\right)+2\le \left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+2$, it follows that

$$\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-4-{\mathsf{\ell }}^{\prime }\ge \left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-4-\mathsf{\ell }\ge \left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-4-\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)-2=2k-7.$$

As $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-4-{\mathsf{\ell }}^{\prime }$ is even, it follows that $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-4-{\mathsf{\ell }}^{\prime }\ge 2k-6$. Therefore, the number of pairwise edge-disjoint copies of $Q_2$ in $H_2^{\prime }$ (and in $H_2$ as well) is at least $\frac{1}{2}(2k-6)=k-3$. By Proposition 4, there is a Ramsey chain of length k in H. □

Next, we illustrate Theorem 5 for some 4-binomial forests of size 10, for which the number of components is 1, 2, or $3=\left(\genfrac{}{}{0pt}{}{4-1}{2}\right)$. Figure 3 shows red–blue colorings of the linear forests $Q_{10}$, $Q_3+Q_7$, and $Q_2+Q_3+Q_5$ of size 10, where four bold edges are red edges and six thin edges are blue edges. For $i=1,2,3,4$, an edge labeled i belongs to the link $G_i$ in a Ramsey chain $G_1,G_2,G_3,G_4$ of length 4 in the linear forest.

4. Binomial Linear Forests with an Intermediate Number of Components

By Theorems 1 and 5, every k-binomial linear forest where $k\ge 4$ with t components, such that either $t=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ or $1\le t\le \left(\genfrac{}{}{0pt}{}{k-1}{2}\right)$, has Ramsey index k. A natural question concerns whether these bounds on t can be improved. As we will see, no such improvement is possible. First, we provide a necessary and sufficient condition for a k-binomial linear forest to have Ramsey index $k-1$.

Theorem 6. 

A k-binomial linear forest H, where $k\ge 4$ has Ramsey index $k-1$ if and only if

(a) 

H contains a subgraph $F=Q_2$, such that $H-E\left(F\right)$ has at most $k-3$ pairwise edge-disjoint copies of $Q_2$ or

(b) 

H contains a subgraph $F\in \{Q_4,2Q_2\}$, such that $H-E\left(F\right)$ has at most $k-4$ pairwise edge-disjoint copies of $Q_2$.

Proof. 

First, we show that if H is a k-binomial linear forest where $k\ge 4$, such that neither (a) nor (b) holds, then $AR\left(H\right)=k$. As $AR\left(H\right)\le k$ by Observation 1, it remains to show that $AR\left(H\right)\ge k$. Let c be a 2-edge coloring of H using the colors 1 and 2. For $i=1,2$, let $H_i$ be the linear forest in H induced by the edges of H colored i. Let $H_i$ have size $m_i$, where $1\le m_1\le m_2$. By Proposition 3, there is a Ramsey chain of length k in H if $H_1\notin \{Q_2,2Q_2,Q_4\}$. Thus, we may assume that $H_1\in \{Q_2,2Q_2,Q_4\}$:

★

If $H_1=Q_2$, let $F=H_1=Q_2$. As (a) does not occur, it follows that $H-E\left(F\right)$ has at least $k-2$ pairwise edge-disjoint copies of $Q_2$. Hence, H has a Ramsey chain of length k by Proposition 4.

★

If $H_1\in \{2Q_2,Q_4\}$, let $F=H_1$. As (b) does not occur, it follows that $H-E\left(F\right)$ has at least $k-3$ pairwise edge-disjoint copies of $Q_2$. Hence, H has a Ramsey chain of length k by Proposition 4.

For the converse, suppose that H is a k-binomial linear forest where $k\ge 4$, such that either (a) or (b) occurs. We show that $AR\left(H\right)=k-1$. As $AR\left(H\right)\ge k-1$ by Corollary 4, it remains to show that $AR\left(H\right)\le k-1$. We consider two cases, according to whether (a) or (b) occurs.

Case 1: $\left(a\right)$ occurs. Let $F=Q_2$ be a subgraph of H, such that $H-E\left(F\right)$ has at most $k-3$ pairwise edge-disjoint copies of $Q_2$. Let c be a 2-edge coloring of H that assigns the color 1 to the two edges of F and the color 2 to all other edges of H. Then $H_1=F=Q_2$ and $H_2=H-E\left(F\right)$. We claim that there is no Ramsey chain of length k with respect to c. Assume, to the contrary, that there is a Ramsey chain $G_1,G_2,\dots ,G_k$ of length k in H. As the size of H is $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$, it follows that $\{G_1,G_2,\dots ,G_k\}$ is a decomposition of H. Necessarily, $G_2=H_1=F=Q_2$, and so, $Q_2\subseteq G_i$ for $3\le i\le k$, which implies that $H-E\left(F\right)$ contains at least $k-2$ pairwise edge-disjoint copies of $Q_2$, which contradicts the fact that $H-E\left(F\right)$ has at most $k-3$ pairwise edge-disjoint copies of $Q_2$. Thus, $AR\left(H\right)\le k-1$, and so, $AR\left(H\right)=k-1$.

Case 2: $\left(b\right)$ occurs. Let $F\in \{Q_4,2Q_2\}$ be a subgraph of H, such that $H-E\left(F\right)$ has at most $k-4$ pairwise edge-disjoint copies of $Q_2$. Let c be a 2-edge coloring of H that assigns the color 1 to the four edges of F and the color 2 to all other edges of H. Then, $H_1=F\in \{Q_4,2Q_2\}$ and $H_2=H-E\left(F\right)$. We claim that there is no Ramsey chain of length k, with respect to c. Assume, to the contrary, that there is a Ramsey chain $G_1,G_2,\dots ,G_k$ of length k in H. As $\{G_1,G_2,\dots ,G_k\}$ is a decomposition of H, it follows that either (i) $H_1$ is decomposed into $G_1=Q_1$ and $G_3\in \{Q_3,Q_2+Q_1\}$ or (ii) $G_4=H_2=F$. If (i) occurs, then $Q_2\subseteq G_i$ for $4\le i\le k$, which implies that $H-E\left(F\right)$ contains at least $k-3$ pairwise edge-disjoint copies of $Q_2$, which contradicts the fact that $H-E\left(F\right)$ has at most $k-4$ pairwise edge-disjoint copies of $Q_2$. If (ii) occurs, then $G_3$ contains a copy of $Q_2$ and $G_i$ contains two edge-disjoint copies of $Q_2$ for $5\le i\le k$, which implies that $H-E\left(F\right)$ contains at least $1+2(k-4)=2k-7>k-4$ pairwise edge-disjoint copies of $Q_2$, which contradicts the fact that $H-E\left(F\right)$ has at most $k-4$ pairwise edge-disjoint copies of $Q_2$. Thus, $AR\left(H\right)\le k-1$, and so, $AR\left(H\right)=k-1$. □

We have now seen for a linear forest H of size m where $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$ and $k\ge 4$ that by Theorems 1, 4, and 5, if (i) $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)<m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$ or (ii) $m=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ and H has t components, where $t=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ or $1\le t\le \left(\genfrac{}{}{0pt}{}{k-1}{2}\right)$, then $AR\left(H\right)=k$. Consequently, it remains only to consider those k-binomial linear forests with an intermediate number t of components, namely, $\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)<t<\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$. Let c be any 2-edge coloring of H using the colors 1 and 2, resulting in the monochromatic subgraphs $H_1$ and $H_2$ of sizes $m_1$ and $m_2$, respectively, where $m_1\le m_2$. By Proposition 3, if $H_1\notin \{Q_2,2Q_2,Q_4\}$, then $A{R}_c\left(H\right)=k$. By Proposition 4 and Theorem 6, if $H_1\in \{Q_2,2Q_2,Q_4\}$, then $A{R}_c\left(H\right)\in \{k,k-1\}$. Furthermore, by Theorem 6, if $H_1=Q_2$, then $A{R}_c\left(H\right)=k$ only when $H_2$ contains at least $k-2$ pairwise edge-disjoint copies of $Q_2$; otherwise, $A{R}_c\left(H\right)=k-1$. Moreover, if $H_1\in \{2Q_2,Q_4\}$, then $A{R}_c\left(H\right)=k$ only when $H_2$ has at least $k-3$ pairwise edge-disjoint copies of $Q_2$; otherwise, $A{R}_c\left(H\right)=k-1$. All of these suggest the need for only considering those 2-edge colorings of H for which $H_1\in \{Q_2,2Q_2,Q_4\}$ and determining whether there is (a) any 2-edge coloring of H where $H_1=Q_2$, such that $H_2$ has fewer than $k-2$ pairwise edge-disjoint copies of $Q_2$ or (b) any 2-edge coloring of H where $H_1\in \{2Q_2,Q_4\}$, such that $H_2$ has fewer than $k-3$ pairwise edge-disjoint copies of $Q_2$. If there is a 2-edge coloring of H resulting in (a) or (b), then $AR\left(H\right)=k-1$; otherwise, $AR\left(H\right)=k$. Therefore, to determine the Ramsey index of a k-binomial linear forest H where $k\ge 4$ with an intermediate number t of components with $\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)<t<\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$, it suffices to study the 2-edge colorings of H, such that $H_1\in \{Q_2,2Q_2,Q_4\}$, such that $H_2=H-E\left(H_1\right)$ possesses the minimum number of pairwise edge-disjoint copies of $Q_2$.

Each k-binomial linear forest H with t of components can be expressed as $Q_{q_1}+Q_{q_2}+\dots +Q_{q_t}$, where $q_1\ge q_2\ge \dots \ge q_t\ge 1$ and ${\sum }_{i=1}^t{q}_i=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$. The maximum number of pairwise edge-disjoint copies of $Q_2$ in H is $s={\sum }_{i=1}^t⌊\frac{q_i}{2}⌋$. Now, let c be any 2-edge coloring of H using the colors 1 and 2 resulting in the monochromatic subgraphs and $H_2$ of sizes $m_1$ and $m_2$, respectively, where $m_1\le m_2$ and $H_1\in \{Q_2,2Q_2,Q_4\}$. First, we make some observations. If $H_1=Q_2$, then the maximum number of pairwise edge-disjoint copies of $Q_2$ in $H_2$ is either $s-1$ or $s-2$; while, if $H_1\in \{Q_4,2Q_2\}$, then the maximum number of pairwise edge-disjoint copies of $Q_2$ in $H_2$ is $s-2$, $s-3$, or $s-4$. We are now prepared to present the following result.

Theorem 7. 

Let $H=Q_{q_1}+Q_{q_2}+\dots +Q_{q_t}$ be a k-binomial linear forest of size ${\sum }_{i=1}^t{q}_i=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ for some integer $k\ge 4$ with t components, where $\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)<t<\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$, where $s={\sum }_{i=1}^t⌊\frac{q_i}{2}⌋$ is the maximum number of pairwise edge-disjoint copies of $Q_2$ in H:

(1)

If $s>k$, then $AR\left(H\right)=k$.

(2)

If $s=k$ and H contains two components of even size 4 or more, then $AR\left(H\right)=k-1$; otherwise, $AR\left(H\right)=k$.

(3)

If $s=k-1$ and H contains at least one component of even size 4 or more, then $AR\left(H\right)=k-1$; otherwise, $AR\left(H\right)=k$.

(4)

If $s\le k-2$, then $AR\left(H\right)=k-1$.

Proof. 

We first verify (1). Suppose that $s>k$. By Observation 1, it suffices to show that $A{R}_c\left(H\right)=k$ for every 2-edge coloring c of H. Let c be a 2-edge coloring of H using the colors 1 and 2. For $i=1,2$, let $H_i$ be the linear forest in H induced by the edges of H colored i where $H_i$ has size $m_i$ and $1\le m_1\le m_2$. By Proposition 3, we may assume that $H_1\in \{Q_2,2Q_2,Q_4\}$. If $H_1=Q_2$, then the number of pairwise edge-disjoint copies of $Q_2$ in $H_2$ is at least $s-2$. As $s-2>k-3$, it follows that $A{R}_c\left(H\right)=k$ by Theorem 6. If $H_1\in \{Q_4,2Q_2\}$, then the number of pairwise edge-disjoint copies of $Q_2$ in $H_2$ is at least $s-4$. As $s-4>k-4$, it follows that $A{R}_c\left(H\right)=k$ by Theorem 6. Therefore, $AR\left(H\right)=k$.

Next, we verify (2). Suppose that $s=k$:

★

First, assume that H contains two components, $Q_x$ and $Q_y$, where $x,y\ge 4$ are both even. Define a 2-edge coloring c of H using the colors 1 and 2, such that $H_1=2Q_2$ and $H_1$ is placed in H in such a way that each of $Q_x$ and $Q_y$ contains a copy of $Q_2$ of $H_1$ and the number of pairwise edge-disjoint copies of $Q_2$ in $H_2$ is $s-4=k-4$. Thus, $A{R}_c\left(H\right)=k-1$ by Theorem 6, and so, $AR\left(H\right)=k-1$.

★

Next, assume that H contains at most one component of even order 4 or more. Let c be any 2-edge coloring of H using the colors 1 and 2. For $i=1,2$, let $H_i$ be the linear forest in H induced by the edges of H colored i where $H_i$ has size $m_i$ and $1\le m_1\le m_2$. By Proposition 3, we may assume that $H_1\in \{Q_2,2Q_2,Q_4\}$. If $H_1=Q_2$, then the number of pairwise edge-disjoint copies of $Q_2$ in $H_2$ is at least $s-2=k-2$, and so, $A{R}_c\left(H\right)=k$ by Theorem 6. If $H_1\in \{Q_4,2Q_2\}$, then, since H contains at most one component of even order 4 or more, any placement of $H_1$ in H produces at least $s-3=k-3$ pairwise edge-disjoint copies of $Q_2$ in $H_2$. It follows by Theorem 6 that $A{R}_c\left(H\right)=k$. Therefore, $AR\left(H\right)=k$.

We now verify (3). Suppose that $s=k-1$:

★

First, assume that H contains at least one component $Q_x$ where $x\ge 4$ is even. Define a 2-edge coloring c of H using the colors 1 and 2, such that $H_1=Q_2$ and $H_1$ is placed in $Q_x$, in such a way that the number of pairwise edge-disjoint copies of $Q_2$ in $H_2$ is $s-2=k-3$. It follows by Theorem 6 that $A{R}_c\left(H\right)=k-1$, and so, $AR\left(H\right)=k-1$.

★

Next, assume that H contains no component of even order 4 or more. Let c be any 2-edge coloring of H using the colors 1 and 2, where $H_i$ is the linear forest of size $m_i$ in H induced by the edges of H colored i and $1\le m_1\le m_2$. By Proposition 3, we may assume that $H_1\in \{Q_2,2Q_2,Q_4\}$. If $H_1=Q_2$, then the number of pairwise edge-disjoint copies of $Q_2$ in $H_2$ is at least $s-1=k-2$, and so, $A{R}_c\left(H\right)=k$ by Theorem 6. If $H_1\in \{Q_4,2Q_2\}$, then, as H contains no component of even order 4 or more, any placement of $H_1$ in H produces at least $s-2=k-3$ pairwise edge-disjoint copies of $Q_2$ in $H_2$. It follows by Theorem 6 that $A{R}_c\left(H\right)=k$. Therefore, $AR\left(H\right)=k$.

Finally, we verify (4). Suppose that $s\le k-2$. Define a 2-edge coloring c of H using the colors 1 and 2, such that $H_1\in \{Q_2,Q_4,2Q_2\}$. Then, the number of pairwise edge-disjoint copies of $Q_2$ in $H_2$ is at least $s-1$. As $s-1\le k-3$, it follows by Theorem 6 that $A{R}_c\left(H\right)=k-1$, and so, $AR\left(H\right)=k-1$. □

We have seen (by Theorems 1 and 5) that if H is a k-binomial linear forest where $k\ge 4$ with t components, whether $t=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ or $1\le t\le \left(\genfrac{}{}{0pt}{}{k-1}{2}\right)$, then $AR\left(H\right)=k$. With the aid of Theorem 7, we are now able to show that these bounds on t cannot be improved.

Theorem 8. 

For every two integers t and k where $\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)<t<\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ and $k\ge 4$, there is a k-binomial linear forest H with t components, such that $AR\left(H\right)=k-1$.

Proof. 

For each integer i with $1\le i\le 2k-2$ where $k\ge 4$, we construct a k-binomial linear forest $F_i$ with $t=\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+i$ components, such that $AR\left(F_i\right)=k-1$. In particular, $F_1$ has $t=\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+1$ components and $F_{2k-2}$ has $t=\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+(2k-2)=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-1$ components.

For an integer $k\ge 4$, let $F_1=Q_x+Q_y+\left[\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)-1\right]Q_1$, where $x=y=k$ if k is even and $x=k+1$ and $y=k-1$ if $k\ge 5$ is odd. Thus, $x,y\ge 4$ are both even. As the size of $F_1$ is $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$, it follows that $F_1$ is a k-binomial linear forest with $t=\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+1$ components. The maximum number of pairwise edge-disjoint copies of $Q_2$ in $F_1$ is $s=⌊\frac{x}{2}⌋+⌊\frac{y}{2}⌋=k$. As $s=k$ and $F_1$ has two components of even size 4 or more, it follows by Theorem 7 that $AR\left(F_1\right)=k-1$.

Next, let $F_2=Q_{x-1}+Q_y+\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)Q_1$. Then, $F_2$ is a k-binomial linear forest with $t=\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+2$ components. The maximum number of pairwise edge-disjoint copies of $Q_2$ in $F_2$ is $s=⌊\frac{x-1}{2}⌋+⌊\frac{y}{2}⌋=k-1$ and $F_2$ has the component $Q_y$ of even size 4 or more. By Theorem 7, $AR\left(F_2\right)=k-1$. Next, let $F_3=Q_{x-2}+Q_y+\left[\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+1\right]Q_1$. Then, $F_3$ is a k-binomial linear forest with $t=\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+3$ components. The maximum number of pairwise edge-disjoint copies of $Q_2$ in $F_3$ is $s=⌊\frac{x-2}{2}⌋+⌊\frac{y}{2}⌋=k-1$ and $F_3$ has the component $Q_y$ of even size 4 or more. By Theorem 7, $AR\left(F_3\right)=k-1$.

Next, let $F_4=Q_{x-2}+Q_{y-1}+\left[\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+2\right]Q_1$. Then, $F_4$ is a k-binomial linear forest with $t=\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+4$ components. The maximum number of pairwise edge-disjoint copies of $Q_2$ in $F_4$ is $s=⌊\frac{x-2}{2}⌋+⌊\frac{y-1}{2}⌋=k-2$. We continue this procedure, reducing the size of components greater than 1 until we arrive at

$$F_{2k-2}=Q_2+Q_1+\left[\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+2k-4\right]Q_1=Q_2+\left[\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+2k-3\right]Q_1=Q_2+\left[\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-2\right]Q_1.$$

Here, $F_{2k-2}$ is a k-binomial linear forest with $t=\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)-1$ components. For each integer i with $4\le i\le 2k-2$, the maximum number of pairwise edge-disjoint copies of $Q_2$ in $F_i$ is $s\le k-2$, and so, $AR\left(F_i\right)=k-1$ by Theorem 7. □

To illustrate Theorem 7, we consider $k=5$ and $\left(\genfrac{}{}{0pt}{}{4}{2}\right)<7\le t\le 14<\left(\genfrac{}{}{0pt}{}{6}{2}\right)$. For $i=1,2,\dots ,8$, we construct a 5-binomial linear forest $F_i$ with $t=\left(\genfrac{}{}{0pt}{}{5-1}{2}\right)+i=6+i$ components, such that $AR\left(F_i\right)=4$:

★

Let $F_1=Q_6+Q_4+5Q_1$, where $t=7$ and $s=5$. As $F_1$ has 2 components $Q_6$ and $Q_4$ of even size 4 or more, $AR\left(F_1\right)=4$ by Theorem 7.

★

Let $F_2=Q_5+Q_4+6Q_1$, where $t=8$ and $s=4$. As $F_2$ has the component $Q_4$ of size 4, it follows by Theorem 7 that $AR\left(F_2\right)=4$.

★

Let $F_3=Q_4+Q_4+7Q_1$, where $t=9$ and $s=4$. As $F_3$ has the component $Q_4$ of size 4, it follows by Theorem 7 that $AR\left(F_3\right)=4$.

★

For $4\le i\le 8$, let

$F_4=Q_4+Q_3+8Q_1$, where $t=10$ and $s=3$,

$F_5=Q_3+Q_3+9Q_1$, where $t=11$ and $s=2$,

$F_6=Q_3+Q_2+10Q_1$, where $t=12$ and $s=2$,

$F_7=Q_2+Q_2+11Q_1$, where $t=13$ and $s=2$, and

$F_8=Q_2+Q_1+12Q_1$, where $t=14$ and $s=1$.

As $s\le 5-2=3$, it follows by Theorem 7 that $AR\left(F_i\right)=4$ for $4\le i\le 8$.

We saw that the 5-binomial linear forest $F_1=Q_6+Q_4+5Q_1$ has 7 components and $AR\left(F_1\right)=4$. This does not imply that every 5-binomial linear forest with 7 components has Ramsey index 4. For example, $F=4Q_3+3Q_1$ is also a 5-binomial linear forest with 7 components and the maximum number of pairwise edge-disjoint copies of $Q_2$ in F is $s=4$. As F has no component of even size 4 or more, it follows by Theorem 7 that $AR\left(F\right)=5$.

5. Closing Comments

From the information obtained on Ramsey chains of linear forest, a question remains, namely that of determining information on Ramsey chains of other familiar classes of graphs. For every graph G of size m that has been investigated where $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$, it has been shown that either $AR\left(G\right)=k$ or $AR\left(G\right)=k-1$. This leads to the following problem:

Problem 1. 

Let G be a graph of size m with $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$ for some positive integer k. Is it true that either $AR\left(G\right)=k$ or $AR\left(G\right)=k-1$?