Ramsey Theorem: R ( 3 , 4 ) ≤ 10 R(3,4)\le 10 R ( 3 , 4 ) ≤ 10 Proof: Why Is The Number Of Friends Be At-least 6 Instead Of 5(by Pigeon Rule, ⌈ 9 / 2 ⌉ = 5 \lceil 9/2\rceil=5 ⌈ 9/2 ⌉ = 5 )?

Ramsey Theorem: $R(3,4)\le 10$ Proof: Why is the number of friends be at-least 6 instead of 5(by pigeon rule, $\lceil 9/2\rceil=5$)?

Ramsey Theory is a branch of mathematics that deals with the conditions under which order must appear. In other words, it studies the conditions under which a certain property must appear in a large structure. One of the most famous results in Ramsey Theory is the Ramsey Theorem, which states that for any given positive integers $r$ and $s$, there exists a positive integer $R(r,s)$ such that if the edges of a complete graph on $R(r,s)$ vertices are colored with two colors, then a complete subgraph of size $r$ with all edges of the same color must exist.

In this article, we will focus on the proof of the Ramsey Theorem for $R(3,4)\le 10$. This result states that if we have a complete graph on 10 vertices, and we color the edges with two colors, then a complete subgraph of size 3 with all edges of the same color must exist.

The Pigeonhole Principle is a fundamental concept in mathematics that states that if $n$ items are put into $m$ containers, with $n>m$, then at least one container must contain more than one item. In other words, if we have more items than containers, then at least one container must be overfilled.

In the context of the Ramsey Theorem, the Pigeonhole Principle is used to prove that a complete subgraph of size 3 with all edges of the same color must exist. To do this, we need to show that if we have a complete graph on 10 vertices, and we color the edges with two colors, then at least one vertex must have at least 5 edges of the same color.

Why is the number of friends be at-least 6 instead of 5(by pigeon rule, $\lceil 9/2\rceil=5$)?

To understand why the number of friends must be at least 6 instead of 5, let's consider the following scenario. Suppose we have a complete graph on 10 vertices, and we color the edges with two colors. Let's say that vertex A has 5 edges of color 1, and 5 edges of color 2. In this case, vertex A has a total of 10 edges, which is the maximum possible number of edges for a vertex in a complete graph.

Now, let's consider the neighbors of vertex A. Since vertex A has 5 edges of color 1, it must have at least 5 neighbors that have edges of color 1. Similarly, since vertex A has 5 edges of color 2, it must have at least 5 neighbors that have edges of color 2.

In other words, vertex A must have at least 10 neighbors, with at least 5 neighbors having edges of color 1, and at least 5 neighbors having edges of color 2. This means that vertex A must have at least 6 friends, with at least 3 friends having edges of color 1, and at least 3 friends having edges of color 2.

To prove that $R(3,4)\le 10$, we need to show that if we have a complete graph on 10 vertices, and we color the edges with two colors, then a complete subgraph of size 3 with all edges of the same color must exist.

Let's assume that this is not the case, and that we have a complete graph on 10 vertices, with edges colored with two colors, such that no complete subgraph of size 3 has all edges of the same color.

In this case, we can consider the neighbors of each vertex. Since each vertex has at least 5 edges of each color, it must have at least 5 neighbors that have edges of each color.

Let's consider the neighbors of vertex A. Since vertex A has at least 5 neighbors that have edges of color 1, and at least 5 neighbors that have edges of color 2, it must have at least 10 neighbors in total.

However, since we have a complete graph on 10 vertices, and each vertex has at least 5 neighbors, it is not possible for vertex A to have at least 10 neighbors. This is a contradiction, and therefore our assumption that no complete subgraph of size 3 has all edges of the same color must be false.

In this article, we have proved that $R(3,4)\le 10$. This result states that if we have a complete graph on 10 vertices, and we color the edges with two colors, then a complete subgraph of size 3 with all edges of the same color must exist.

We have used the Pigeonhole Principle to prove this result, and have shown that if we have a complete graph on 10 vertices, with edges colored with two colors, then at least one vertex must have at least 5 edges of each color.

This result has important implications for the study of Ramsey Theory, and has been used to prove many other results in this field.
Ramsey Theorem: $R(3,4)\le 10$ Proof: Q&A

In our previous article, we proved that $R(3,4)\le 10$. This result states that if we have a complete graph on 10 vertices, and we color the edges with two colors, then a complete subgraph of size 3 with all edges of the same color must exist.

In this article, we will answer some of the most frequently asked questions about the Ramsey Theorem and its proof. We will also provide additional information and examples to help clarify the concepts.

Q: What is the Ramsey Theorem?

A: The Ramsey Theorem is a result in mathematics that states that for any given positive integers $r$ and $s$, there exists a positive integer $R(r,s)$ such that if the edges of a complete graph on $R(r,s)$ vertices are colored with two colors, then a complete subgraph of size $r$ with all edges of the same color must exist.

Q: What is the significance of the Ramsey Theorem?

A: The Ramsey Theorem has many significant implications in mathematics and computer science. It has been used to prove many other results in Ramsey Theory, and has applications in graph theory, combinatorics, and computer science.

Q: How is the Ramsey Theorem related to the Pigeonhole Principle?

A: The Ramsey Theorem is closely related to the Pigeonhole Principle. The Pigeonhole Principle states that if $n$ items are put into $m$ containers, with $n>m$, then at least one container must contain more than one item. In the context of the Ramsey Theorem, the Pigeonhole Principle is used to prove that a complete subgraph of size 3 with all edges of the same color must exist.

Q: What is the difference between $R(3,4)$ and $R(4,3)$?

A: $R(3,4)$ and $R(4,3)$ are two different values of the Ramsey Theorem. $R(3,4)$ is the smallest number of vertices such that if the edges of a complete graph on $R(3,4)$ vertices are colored with two colors, then a complete subgraph of size 3 with all edges of the same color must exist. $R(4,3)$ is the smallest number of vertices such that if the edges of a complete graph on $R(4,3)$ vertices are colored with two colors, then a complete subgraph of size 4 with all edges of the same color must exist.

Q: Can you provide an example of a complete graph on 10 vertices with edges colored with two colors, such that no complete subgraph of size 3 has all edges of the same color?

A: Unfortunately, it is not possible to provide an example of a complete graph on 10 vertices with edges colored with two colors, such that no complete subgraph of size 3 has all edges of the same color. This is because the Ramsey Theorem states that such a graph does not exist.

Q: What are some of the applications of the Ramsey Theorem?

A: The Ramsey Theorem has many applications in mathematics computer science. Some of the applications include:

• Graph theory: The Ramsey Theorem has been used to study the properties of graphs, such as the existence of cliques and the distribution of edges.
• Combinatorics: The Ramsey Theorem has been used to study the properties of combinatorial objects, such as permutations and combinations.
• Computer science: The Ramsey Theorem has been used to study the properties of algorithms and data structures, such as the existence of efficient algorithms for solving problems.

In this article, we have answered some of the most frequently asked questions about the Ramsey Theorem and its proof. We have also provided additional information and examples to help clarify the concepts. The Ramsey Theorem is a fundamental result in mathematics that has many significant implications in mathematics and computer science.