# Ramsey Numbers -- Breakthrough claimed

#### WBahn

Joined Mar 31, 2012
29,469
https://www.iflscience.com/rare-bre...ns-your-parties-just-got-more-efficient-68271

When you read it, be aware that the description of the problem given is somewhat misleading: "Imagine you’re hosting a mixer, and you want to make sure there’s a good balance of guests who know each other and guests who are strangers. What is the minimum number of people you need to invite to ensure there will be at least one group of three who either all know each other or are all strangers?"

The first sentence implies that you want a balance of the two possibilities, which would mean that you want to have a mix of both groups that know each other and groups that are strangers at the party. But the actual problem, as accurately reflected in the second sentence, is to find the minimum number of attendees that ensures that there is at least one group of three people that are either all mutual acquaintances or all mutual strangers; if you have one type of group, you don't care whether you have the other.

So, if you invite two of your friends to a party, then your party has a group of three people (including you) that all know each other and you are happy, even though there is no "balance" between guess that know each other and guests that don't. But this requires you to cherry pick the attendees, since if you sent out invites to a mailing list that included everyone in a large company, let's say, and only two people showed up, then it's possible that you know one of them and not the other, and hence there is no group of three that are either all mutual acquaintances or all mutual strangers. The question asks what the minimum number of attendees is such that you are guaranteed, not matter how carefully you might try to avoid it, that you WILL have at least one group of k people (k being three is this example) that are either all mutual acquaintances or all mutual strangers.

The actual problem is from graph theory in the field of combinatorics. The party-problem is merely a way of phrasing an example.