There are n new students in a class. Among any three of them, there ex

HoneyLemon

I am sure you will never get such a question in the actuals, but it is always good to understand different concepts. Not an easy one by any standard.
But all the above solutions are way off the mark.

Two conditions
1) Pick any 3 of them and you will have two who know each other.
So there will be groups of people who know one another within the group. Because if you have three isolated groups, you can choose one from each group and none of three would know each other.
2) Pick any 4 of them and you will have two who do not know each other.
This restricts the number of people in any one group to 3
Because if there are 4 in the group, you can pick up these 4 and all would know each other.

With these two conditions, the immediate answer could be 6 : 3 in each group
ABC and DEF

But we are looking for max number, so we can think of interconnected groups.
Say, in each new group, two remain the same and one is changed. But we cannot have more than 2 completely isolated groups.

So groups can be
ABC
BCD
CDE
DEF
EFG
FGH
GHA

Now, we don’t go for next group GHI, that is we do not introduce new member I.
Because then we will have 3 isolated groups : ABC, DEF and GHI. The moment we pick up one from each of these 3 groups, we will not have two people known to each other. Example ADG or AEH and so on.

Now, we can get back to our two conditions and see if they are fulfilled.
(1) Pick any three : Only two groups will be completely isolated. So the third student picked up will be related to at least one of these two groups.
(2) Pick any four : As we have groups of only 3 people who are known to one another, the fourth person added will not know at least one of them. Example ABCD - A and D are not known to each other. OR ABCF - Again F does not know A, B or C.

Total students - A, B, C, D, E, F, G and H, so 8 students. You can also draw a circle and solve, but this is much easier to grasp this way.

B