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This is my first puzzle so I hope everything is OK.

You are a principal of a small middle school. Koro and Irina are two teachers responsible for teaching a senior class which has 13 female and 15 male students.

Now, this class has a significant number of problematic students so you want to assign a teacher that could help them bond and build a socially better in-class environment. You ask both of them to join you in a meeting in order to decide who has more potential for the task. Below is your meeting chat.

You: So, about this class, do you guys know how many friend groups there are?

Koro: I am afraid not.

Irina: I am not entirely sure.

You: Oh, hmm, I wanted to compare them with a junior class I teach. Anyway, could you at least tell me the size of the largest friend group you remember?

Koro: Oh, I believe there is a group of 5 male delinquents, they seem to be pretty close.

Irina: I must disagree. I don't think that class has spent enough time together to have a friend group of that size.

You: That is actually what I wanted to discuss.

Koro: ... (Doesn't think the class is that poor socially so he is a little confused.)

Irina: (Realizes this) Okay, it was mainly the size I disagreed with, Koro, not the bondedness. I mean, take any 4 students, for example, I am sure 2 or 3 of them would be friends, but not even 4! Let alone a group of 5...

Koro: Even if what you said was true for some of the class, I still think you are mistaken about the delinquents, Irina. I always see them together!

Now, you find both of them a little inadequate for the issue, but you must pick one, so you decide to pick the one who isn't wrong for sure.

Who is handed the task?

Hints/Notes

- A friend group is a group of people who are all friends.
- Friendships are mutual.
- You might not use all the information to reach a conclusion.

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    $\begingroup$ It's not clear what you're asking here. Do you want to know In a group of 13 female and 15 male people, is it more likely that there is a group of 5 male friends or that every group of 4 students has at least two of them being friends?? $\endgroup$ – Ian MacDonald Aug 24 '16 at 14:00
  • $\begingroup$ @IanMacDonald Like I asked, I was trying to ask which one of them is less likely wrong. As the answer below says, K may be wrong or right whereas I is wrong for sure. $\endgroup$ – Sora Aug 24 '16 at 14:03
  • $\begingroup$ This is a stretch maybe, but do the names refer to Assassination Classroom? ;) $\endgroup$ – Max Li Aug 26 '16 at 18:14
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    $\begingroup$ @MaxLi yes they are :) also the size of the class $\endgroup$ – Sora Aug 27 '16 at 10:57
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We should (maaaybe, see below) choose

K rather than I

because

I's account of affairs in the class is internally inconsistent.

That's because

she claims that given any 4 pupils there will be at least one pair of friends, but there is no group of 4 who are all friends

but

these can't both be true in a class of 28. In fact, they can't be true in a class of 18, because the Ramsey number R(4,4) is 18. (The mathematicianese way of saying this is: given any graph with 18 vertices, either the graph or its complement contains a complete graph with 4 vertices as a subgraph.)

The "maaaybe" above is because of course

it's actually not at all clear to me that "reject anyone who has definitely said at least one wrong thing" is a good strategy.

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  • $\begingroup$ Yes, also I says there is no group of 5 friends but also no group with 4 people who aren't friends which is also against R(5,4) = 25. Of course this isn't a good strategy, I just wanted people to find which one of them is inconsistent/wrong. $\endgroup$ – Sora Aug 24 '16 at 14:05
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    $\begingroup$ Yes, even if she hadn't added "not even 4" she would have been inconsistent, but not quite so drastically. $\endgroup$ – Gareth McCaughan Aug 24 '16 at 14:57
  • $\begingroup$ Doesn't I object to there being a friendly group of 5 male delinquents? Given that there are only 15 male students, this does not lead to a contradiction. $\endgroup$ – Shagnik Aug 26 '16 at 15:41
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    $\begingroup$ That's not my understanding of what I is saying. "I don't think that class has spent enough time together to have a friend group of that size." (not "... a friend group of that size all of whom are the same sex" or anything); "take any 4 students, for example, I am sure 2 or 3 of them would be friends, but not even 4! Let alone a group of 5..." (again, nothing about the sex of the students). $\endgroup$ – Gareth McCaughan Aug 26 '16 at 15:50
  • $\begingroup$ @Shagnik gareth explained it pretty well, I dont think there's a gender-specific claim made by I. $\endgroup$ – Sora Aug 27 '16 at 10:57

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