How can you prove among any group of 6 people there are either 3 mutual friends or 3 mutual strangers?

(My reasoning) Any person either has (atleast 3 friends) OR (atleast 3 enemies). This is true because each person can be friend\enemy with five others and by the pigeonhole principle atleast one of the two holes (friend , enemy ) must contain 3 or more people.

How many people would you need at a party to guarantee that at least three individuals know each other or that at least three do not know each other?

If the 3 people are not connected by any red lines, then all 3 of them are connected by blue lines forming a blue triangle which represents 3 people not knowing each other. Thus in a group of 6 people there will always be 3 people who know each other or 3 people who do not know each other.

What is the minimum number of people a party should invite so that at least three are mutual friends and at least three are mutual strangers?

six people
The theorem says: In any party of six people either at least three of them are (pairwise) mutual strangers or at least three of them are (pairwise) mutual acquaintances.

How is Ramsey number calculated?

A Ramsey Number, written as n = R(r, b), is the smallest integer n such that the 2-colored graph Kn, using the colors red and blue for edges, implies a red monochromatic subgraph Kr or a blue monochromatic subgraph Kb. [1] 5 Page 6 There are a couple things to note about this definition.

How many friends must you have to ensure that at least five of them will have birthdays on the same month use extended pigeonhole principle?

49 friends
Extended pigeonhole principle. ∴ 49 friends should be their to guarantee that at-least five of them must have birthday in a same month of year.

How many people do you need to gather to be sure that at least two of them have birthdays in the same month?

In a room of just 23 people there’s a 50-50 chance of at least two people having the same birthday. In a room of 75 there’s a 99.9\% chance of at least two people matching.

How large must a group of people be in order to guarantee that there are at least two people in the group whose birthdays fall in the same month?

In its simplest form, applied to the context of your question, the pigeonhole principle states that for m=12 months, if there are n≥13 people in a group, then there is guaranteed to be a month in which at least two people’s birthdays occur.

How many guests do I have to invite to my Christmas party to be sure there will be at least 3 mutual friends or 3 mutual strangers?

We can now see that inviting six people is sufficient to guarantee at least three mutual friends or at least three mutual strangers.

Why are Ramsey numbers so hard to calculate?

Ramsey numbers are hard to calculate because the complexity of a graph increases dramatically as you add vertices. For a graph with six vertices and two colors, you can run through all the possibilities by hand. But for a graph with 40 vertices, there are 2780 ways of applying two colors.

What is Ramsey math theory?

Ramsey theory is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a specific size. This paper will explore some basic definitions of and history behind Ramsey theory, but will focus on a subsection of Ramsey theory known as Ramsey numbers.

How many friends you must have to guarantee that at least 7 of them will have birthdays in the same month *?

Therefore 73 persons will guarantee that at least 7 persons have their birthday in the same month.

What is the minimum number of person in a group necessary to guarantee that at least 4 of them were born in same day of the week?

Using the generalized pigeonhole principle, we need 3 ⋅ 12+1 people to force 4 of them to be born in the same month. At least 2 people were born exactly one week apart. Again, you cannot force this property. For example, everyone might be born on the same day of the year.